id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-16000 | Skewness of Gamma Distribution | Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the Gamma distribution.
Then the skewness $\gamma_1$ of $X$ is given by:
:$\gamma_1 = \dfrac 2 {\sqrt \alpha}$ | From Expectation of Power of Gamma Distribution, we have:
:$\expect {X^n} = \dfrac {\alpha^{\overline n} } {\beta^n}$
where $\alpha^{\overline n}$ denotes the $n$th rising factorial of $\alpha$.
Hence:
{{begin-eqn}}
{{eqn | l = \gamma_1
| r = \expect {\paren {\dfrac {X - \mu} \sigma}^3}
| c = {{Defof|Skewne... | Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the [[Definition:Gamma Distribution|Gamma distribution]].
Then the [[Definition:Skewness|skewness]] $\gamma_1$ of $X$ is given by:
:$\gamma_1 = \dfrac 2 {\sqrt \alpha}$ | From [[Expectation of Power of Gamma Distribution]], we have:
:$\expect {X^n} = \dfrac {\alpha^{\overline n} } {\beta^n}$
where $\alpha^{\overline n}$ denotes the [[Definition:Rising Factorial|$n$th rising factorial]] of $\alpha$.
Hence:
{{begin-eqn}}
{{eqn | l = \gamma_1
| r = \expect {\paren {\dfrac {X - \... | Skewness of Gamma Distribution/Proof 2 | https://proofwiki.org/wiki/Skewness_of_Gamma_Distribution | https://proofwiki.org/wiki/Skewness_of_Gamma_Distribution/Proof_2 | [
"Gamma Distribution",
"Skewness",
"Skewness of Gamma Distribution"
] | [
"Definition:Gamma Distribution",
"Definition:Skewness"
] | [
"Expectation of Power of Gamma Distribution",
"Definition:Rising Factorial",
"Binomial Theorem/Examples/Cube of Difference",
"Expectation is Linear",
"Variance of Gamma Distribution",
"Expectation of Power of Gamma Distribution"
] |
proofwiki-16001 | Excess Kurtosis of Gamma Distribution | Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the Gamma distribution.
Then the excess kurtosis $\gamma_2$ of $X$ is given by:
:$\gamma_2 = \dfrac 6 \alpha$ | From the definition of excess kurtosis, we have:
:$\gamma_2 = \expect {\paren {\dfrac {X - \mu} \sigma}^4} - 3$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
By Expectation of Gamma Distribution, we have:
:$\mu = \dfrac \alpha \beta$
By Variance of Gamma Distribution, we have:... | Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the [[Definition:Gamma Distribution|Gamma distribution]].
Then the [[Definition:Excess Kurtosis|excess kurtosis]] $\gamma_2$ of $X$ is given by:
:$\gamma_2 = \dfrac 6 \alpha$ | From the definition of [[Definition:Excess Kurtosis|excess kurtosis]], we have:
:$\gamma_2 = \expect {\paren {\dfrac {X - \mu} \sigma}^4} - 3$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
By [[Expectation of Gamma... | Excess Kurtosis of Gamma Distribution/Proof 1 | https://proofwiki.org/wiki/Excess_Kurtosis_of_Gamma_Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Gamma_Distribution/Proof_1 | [
"Kurtosis",
"Gamma Distribution",
"Excess Kurtosis of Gamma Distribution"
] | [
"Definition:Gamma Distribution",
"Definition:Excess Kurtosis"
] | [
"Definition:Excess Kurtosis",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Gamma Distribution",
"Variance of Gamma Distribution",
"Kurtosis in terms of Non-Central Moments",
"Skewness of Gamma Distribution",
"Variance of Gamma Distribution",
"Definition:Moment Generatin... |
proofwiki-16002 | Excess Kurtosis of Gamma Distribution | Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the Gamma distribution.
Then the excess kurtosis $\gamma_2$ of $X$ is given by:
:$\gamma_2 = \dfrac 6 \alpha$ | From the definition of excess kurtosis, we have:
:$\gamma_2 = \expect {\paren {\dfrac {X - \mu} \sigma}^4} - 3$
where:
:$\mu = \expect X$ is the expectation of $X$
:$\sigma = \sqrt {\var X}$ is the standard deviation of $X$.
By Expectation of Gamma Distribution, we have:
:$\mu = \dfrac \alpha \beta$
By Variance of G... | Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the [[Definition:Gamma Distribution|Gamma distribution]].
Then the [[Definition:Excess Kurtosis|excess kurtosis]] $\gamma_2$ of $X$ is given by:
:$\gamma_2 = \dfrac 6 \alpha$ | From the definition of [[Definition:Excess Kurtosis|excess kurtosis]], we have:
:$\gamma_2 = \expect {\paren {\dfrac {X - \mu} \sigma}^4} - 3$
where:
:$\mu = \expect X$ is the [[Definition:Expectation|expectation]] of $X$
:$\sigma = \sqrt {\var X}$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.... | Excess Kurtosis of Gamma Distribution/Proof 2 | https://proofwiki.org/wiki/Excess_Kurtosis_of_Gamma_Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Gamma_Distribution/Proof_2 | [
"Kurtosis",
"Gamma Distribution",
"Excess Kurtosis of Gamma Distribution"
] | [
"Definition:Gamma Distribution",
"Definition:Excess Kurtosis"
] | [
"Definition:Excess Kurtosis",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Gamma Distribution",
"Variance of Gamma Distribution",
"Expectation of Power of Gamma Distribution",
"Definition:Rising Factorial",
"Kurtosis in terms of Non-Central Moments",
"Expectation of Gam... |
proofwiki-16003 | Excess Kurtosis of Binomial Distribution | Let $X$ be a discrete random variable with a binomial distribution with parameters $n$ and $p$ for some $n \in \N$ and $0 \le p \le 1$:
Then the excess kurtosis $\gamma_2$ of $X$ is given by:
:$\gamma_2 = \dfrac {1 - 6 p q} {n p q}$
where $q = 1 - p$. | From the definition of excess kurtosis, we have:
:$\gamma_2 = \expect {\paren {\dfrac {X - \mu} \sigma}^4} - 3$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
By Expectation of Binomial Distribution, we have:
:$\mu = n p$
By Variance of Binomial Distribution, we have:
:$\sigma... | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] with a [[Definition:Binomial Distribution|binomial distribution with parameters $n$ and $p$]] for some $n \in \N$ and $0 \le p \le 1$:
Then the [[Definition:Excess Kurtosis|excess kurtosis]] $\gamma_2$ of $X$ is given by:
:$\gamma_2 = \dfr... | From the definition of [[Definition:Excess Kurtosis|excess kurtosis]], we have:
:$\gamma_2 = \expect {\paren {\dfrac {X - \mu} \sigma}^4} - 3$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
By [[Expectation of Binom... | Excess Kurtosis of Binomial Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Binomial_Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Binomial_Distribution | [
"Kurtosis",
"Binomial Distribution"
] | [
"Definition:Random Variable/Discrete",
"Definition:Binomial Distribution",
"Definition:Excess Kurtosis"
] | [
"Definition:Excess Kurtosis",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Binomial Distribution",
"Variance of Binomial Distribution",
"Kurtosis in terms of Non-Central Moments",
"Skewness of Binomial Distribution",
"Variance of Binomial Distribution",
"Definition:Mome... |
proofwiki-16004 | Excess Kurtosis of Beta Distribution | Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname{Beta}$ is the Beta distribution.
Then the excess kurtosis $\gamma_2$ of $X$ is given by:
:$\gamma_2 = \dfrac {6 \paren {\paren {\alpha - \beta}^2 \paren {\alpha + \beta + 1} - \alpha \beta \paren {\alpha + \beta + 2} } } {\alpha \be... | From Kurtosis in terms of Non-Central Moments, we have:
:$\gamma_2 = \dfrac {\expect {X^4} - 4 \mu \expect {X^3} + 6 \mu^2 \expect {X^2} - 3 \mu^4} {\sigma^4} - 3$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
We have, by Expectation of Beta Distribution:
:$\expect X = \dfrac {... | Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname{Beta}$ is the [[Definition:Beta Distribution|Beta distribution]].
Then the [[Definition:Excess Kurtosis|excess kurtosis]] $\gamma_2$ of $X$ is given by:
:$\gamma_2 = \dfrac {6 \paren {\paren {\alpha - \beta}^2 \paren {\alpha + \be... | From [[Kurtosis in terms of Non-Central Moments]], we have:
:$\gamma_2 = \dfrac {\expect {X^4} - 4 \mu \expect {X^3} + 6 \mu^2 \expect {X^2} - 3 \mu^4} {\sigma^4} - 3$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
... | Excess Kurtosis of Beta Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Beta_Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Beta_Distribution | [
"Excess Kurtosis of Beta Distribution",
"Kurtosis",
"Beta Distribution"
] | [
"Definition:Beta Distribution",
"Definition:Excess Kurtosis"
] | [
"Kurtosis in terms of Non-Central Moments",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Beta Distribution",
"Variance of Beta Distribution",
"Raw Moment of Beta Distribution",
"Addition of Fractions",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
... |
proofwiki-16005 | Skewness of Beta Distribution | Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname {Beta}$ denotes the Beta distribution.
Then the skewness $\gamma_1$ of $X$ is given by:
:$\gamma_1 = \dfrac {2 \paren {\beta - \alpha} \sqrt {\alpha + \beta + 1} } {\paren {\alpha + \beta + 2} \sqrt {\alpha \beta} }$ | From Skewness in terms of Non-Central Moments:
:$\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
We have, by Expectation of Beta Distribution:
:$\expect X = \dfrac {\alpha} {\alpha + \beta}$
By Variance of Be... | Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname {Beta}$ denotes the [[Definition:Beta Distribution|Beta distribution]].
Then the [[Definition:Skewness|skewness]] $\gamma_1$ of $X$ is given by:
:$\gamma_1 = \dfrac {2 \paren {\beta - \alpha} \sqrt {\alpha + \beta + 1} } {\paren ... | From [[Skewness in terms of Non-Central Moments]]:
:$\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
We have, by [[Expectation of Beta Distribut... | Skewness of Beta Distribution | https://proofwiki.org/wiki/Skewness_of_Beta_Distribution | https://proofwiki.org/wiki/Skewness_of_Beta_Distribution | [
"Skewness",
"Beta Distribution"
] | [
"Definition:Beta Distribution",
"Definition:Skewness"
] | [
"Skewness in terms of Non-Central Moments",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Beta Distribution",
"Variance of Beta Distribution",
"Raw Moment of Beta Distribution",
"Addition of Fractions",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
... |
proofwiki-16006 | Skewness of Exponential Distribution | Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{> 0}$
Then the skewness $\gamma_1$ of $X$ is equal to $2$. | From Skewness in terms of Non-Central Moments, we have:
:$\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
By Expectation of Exponential Distribution we have:
:$\mu = \beta$
By Variance of Exponential Distrib... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] of the [[Definition:Exponential Distribution|exponential distribution]] with parameter $\beta$ for some $\beta \in \R_{> 0}$
Then the [[Definition:Skewness|skewness]] $\gamma_1$ of $X$ is equal to $2$. | From [[Skewness in terms of Non-Central Moments]], we have:
:$\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
By [[Expectation of Exponential Dis... | Skewness of Exponential Distribution | https://proofwiki.org/wiki/Skewness_of_Exponential_Distribution | https://proofwiki.org/wiki/Skewness_of_Exponential_Distribution | [
"Skewness",
"Exponential Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Exponential Distribution",
"Definition:Skewness"
] | [
"Skewness in terms of Non-Central Moments",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Exponential Distribution",
"Variance of Exponential Distribution",
"Raw Moment of Exponential Distribution"
] |
proofwiki-16007 | Excess Kurtosis of Exponential Distribution | Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{> 0}$.
Then the excess kurtosis $\gamma_2$ of $X$ is equal to $6$. | From the definition of excess kurtosis, we have:
:$\gamma_2 = \expect {\paren {\dfrac {X - \mu} \sigma}^4} - 3$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
By Expectation of Exponential Distribution we have:
:$\mu = \beta$
By Variance of Exponential Distribution we have:
:... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] of the [[Definition:Exponential Distribution|exponential distribution]] with parameter $\beta$ for some $\beta \in \R_{> 0}$.
Then the [[Definition:Excess Kurtosis|excess kurtosis]] $\gamma_2$ of $X$ is equal to $6$. | From the definition of [[Definition:Excess Kurtosis|excess kurtosis]], we have:
:$\gamma_2 = \expect {\paren {\dfrac {X - \mu} \sigma}^4} - 3$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
By [[Expectation of Expon... | Excess Kurtosis of Exponential Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Exponential_Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Exponential_Distribution | [
"Kurtosis",
"Exponential Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Exponential Distribution",
"Definition:Excess Kurtosis"
] | [
"Definition:Excess Kurtosis",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Exponential Distribution",
"Variance of Exponential Distribution",
"Kurtosis in terms of Non-Central Moments",
"Raw Moment of Exponential Distribution",
"Category:Kurtosis",
"Category:Exponential... |
proofwiki-16008 | Excess Kurtosis of Poisson Distribution | Let $X$ be a discrete random variable with a Poisson distribution with parameter $\lambda$.
Then the excess kurtosis $\gamma_2$ of $X$ is given by:
:$\gamma_2 = \dfrac 1 \lambda$ | From the definition of excess kurtosis, we have:
:$\gamma_2 = \expect {\paren {\dfrac {X - \mu} \sigma}^4} - 3$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
By Expectation of Poisson Distribution, we have:
:$\mu = \lambda$
By Variance of Poisson Distribution, we have:
:$\sig... | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] with a [[Definition:Poisson Distribution|Poisson distribution with parameter $\lambda$]].
Then the [[Definition:Excess Kurtosis|excess kurtosis]] $\gamma_2$ of $X$ is given by:
:$\gamma_2 = \dfrac 1 \lambda$ | From the definition of [[Definition:Excess Kurtosis|excess kurtosis]], we have:
:$\gamma_2 = \expect {\paren {\dfrac {X - \mu} \sigma}^4} - 3$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
By [[Expectation of Poiss... | Excess Kurtosis of Poisson Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Poisson_Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Poisson_Distribution | [
"Poisson Distribution",
"Kurtosis"
] | [
"Definition:Random Variable/Discrete",
"Definition:Poisson Distribution",
"Definition:Excess Kurtosis"
] | [
"Definition:Excess Kurtosis",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Poisson Distribution",
"Variance of Poisson Distribution",
"Kurtosis in terms of Non-Central Moments",
"Skewness of Poisson Distribution",
"Variance of Poisson Distribution/Proof 1",
"Definition:... |
proofwiki-16009 | Farey Sequence has Convergent Subsequences for all x in Closed Unit Interval | Consider the Farey sequence:
:$\sequence {a_n} = \dfrac 1 2, \dfrac 1 3, \dfrac 2 3, \dfrac 1 4, \dfrac 2 4, \dfrac 3 4, \dfrac 1 5, \dfrac 2 5, \dfrac 3 5, \dfrac 4 5, \dfrac 1 6, \ldots$
Every element of the closed real interval $\closedint 0 1$ is the limit of a subsequence of $\sequence {a_n}$. | We have that every rational number $\dfrac p q$ between $0$ and $1$ occurs infinitely often in $\sequence {a_n}$:
:$\dfrac p q, \dfrac {2 p} {2 q}, \dfrac {3 p} {3 q}, \ldots$
Let $x \in \closedint 0 1$.
From Between two Real Numbers exists Rational Number, a term $a_{n_1}$ of $\sequence {a_n}$ can be found such that:
... | Consider the [[Definition:Farey Sequence|Farey sequence]]:
:$\sequence {a_n} = \dfrac 1 2, \dfrac 1 3, \dfrac 2 3, \dfrac 1 4, \dfrac 2 4, \dfrac 3 4, \dfrac 1 5, \dfrac 2 5, \dfrac 3 5, \dfrac 4 5, \dfrac 1 6, \ldots$
Every [[Definition:Element|element]] of the [[Definition:Closed Real Interval|closed real interval]... | We have that every [[Definition:Rational Number|rational number]] $\dfrac p q$ between $0$ and $1$ occurs [[Definition:Infinite Set|infinitely often]] in $\sequence {a_n}$:
:$\dfrac p q, \dfrac {2 p} {2 q}, \dfrac {3 p} {3 q}, \ldots$
Let $x \in \closedint 0 1$.
From [[Between two Real Numbers exists Rational Number... | Farey Sequence has Convergent Subsequences for all x in Closed Unit Interval | https://proofwiki.org/wiki/Farey_Sequence_has_Convergent_Subsequences_for_all_x_in_Closed_Unit_Interval | https://proofwiki.org/wiki/Farey_Sequence_has_Convergent_Subsequences_for_all_x_in_Closed_Unit_Interval | [
"Farey Sequences",
"Subsequences",
"Limits of Sequences"
] | [
"Definition:Farey Sequence",
"Definition:Element",
"Definition:Real Interval/Closed",
"Definition:Limit of Sequence/Real Numbers",
"Definition:Subsequence"
] | [
"Definition:Rational Number",
"Definition:Infinite Set",
"Between two Real Numbers exists Rational Number",
"Definition:Term of Sequence",
"Definition:Term of Sequence",
"Definition:Subsequence",
"Squeeze Theorem/Sequences/Real Numbers"
] |
proofwiki-16010 | Expectation of Chi-Squared Distribution | Let $n$ be a strictly positive integer.
Let $X \sim \chi_n^2$ where $\chi_n^2$ is the chi-squared distribution with $n$ degrees of freedom.
Then the expectation of $X$ is given by:
:$\expect X = n$ | {{begin-eqn}}
{{eqn | l = \expect X
| r = \prod_{k \mathop = 0}^0 \paren {n + 2 k}
| c = Raw Moment of Chi-Squared Distribution
}}
{{eqn | r = n
}}
{{end-eqn}}
{{qed}} | Let $n$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $X \sim \chi_n^2$ where $\chi_n^2$ is the [[Definition:Chi-Squared Distribution|chi-squared distribution]] with $n$ degrees of freedom.
Then the [[Definition:Expectation|expectation]] of $X$ is given by:
:$\expect X = n$ | {{begin-eqn}}
{{eqn | l = \expect X
| r = \prod_{k \mathop = 0}^0 \paren {n + 2 k}
| c = [[Raw Moment of Chi-Squared Distribution]]
}}
{{eqn | r = n
}}
{{end-eqn}}
{{qed}} | Expectation of Chi-Squared Distribution | https://proofwiki.org/wiki/Expectation_of_Chi-Squared_Distribution | https://proofwiki.org/wiki/Expectation_of_Chi-Squared_Distribution | [
"Expectation",
"Chi-Squared Distribution"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Chi-Squared Distribution",
"Definition:Expectation"
] | [
"Raw Moment of Chi-Squared Distribution"
] |
proofwiki-16011 | Variance of Chi-Squared Distribution | Let $n$ be a strictly positive integer.
Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.
Then the variance of $X$ is given by:
:$\var X = 2 n$ | By Variance as Expectation of Square minus Square of Expectation, we have:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
By Expectation of Chi-Squared Distribution, we have:
:$\expect X = n$
We also have:
{{begin-eqn}}
{{eqn | l = \expect {X^2}
| r = \prod_{k \mathop = 0}^1 \paren {n + 2 k}
| c = Raw ... | Let $n$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $X \sim \chi^2_n$ where $\chi^2_n$ is the [[Definition:Chi-Squared Distribution|chi-squared distribution]] with $n$ degrees of freedom.
Then the [[Definition:Variance|variance]] of $X$ is given by:
:$\var X = 2 n$ | By [[Variance as Expectation of Square minus Square of Expectation]], we have:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
By [[Expectation of Chi-Squared Distribution]], we have:
:$\expect X = n$
We also have:
{{begin-eqn}}
{{eqn | l = \expect {X^2}
| r = \prod_{k \mathop = 0}^1 \paren {n + 2 k}
... | Variance of Chi-Squared Distribution | https://proofwiki.org/wiki/Variance_of_Chi-Squared_Distribution | https://proofwiki.org/wiki/Variance_of_Chi-Squared_Distribution | [
"Variance",
"Chi-Squared Distribution"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Chi-Squared Distribution",
"Definition:Variance"
] | [
"Variance as Expectation of Square minus Square of Expectation",
"Expectation of Chi-Squared Distribution",
"Raw Moment of Chi-Squared Distribution"
] |
proofwiki-16012 | Skewness of Chi-Squared Distribution | Let $n$ be a strictly positive integer.
Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.
Then the skewness $\gamma_1$ of $X$ is given by:
:$\gamma_1 = \sqrt{\dfrac 8 n}$ | From Skewness in terms of Non-Central Moments, we have:
:$\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
By Expectation of Chi-Squared Distribution we have:
:$\mu = n$
By Variance of Chi-Squared Distribution... | Let $n$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $X \sim \chi^2_n$ where $\chi^2_n$ is the [[Definition:Chi-Squared Distribution|chi-squared distribution]] with $n$ degrees of freedom.
Then the [[Definition:Skewness|skewness]] $\gamma_1$ of $X$ is given by:
:$\gamma_1 = \sqrt{\... | From [[Skewness in terms of Non-Central Moments]], we have:
:$\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
By [[Expectation of Chi-Squared Dis... | Skewness of Chi-Squared Distribution | https://proofwiki.org/wiki/Skewness_of_Chi-Squared_Distribution | https://proofwiki.org/wiki/Skewness_of_Chi-Squared_Distribution | [
"Skewness",
"Chi-Squared Distribution"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Chi-Squared Distribution",
"Definition:Skewness"
] | [
"Skewness in terms of Non-Central Moments",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Chi-Squared Distribution",
"Variance of Chi-Squared Distribution",
"Raw Moment of Chi-Squared Distribution",
"Category:Skewness",
"Category:Chi-Squared Distribution"
] |
proofwiki-16013 | Excess Kurtosis of Chi-Squared Distribution | Let $n$ be a strictly positive integer.
Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.
Then the excess kurtosis $\gamma_2$ of $X$ is given by:
:$\gamma_2 = \dfrac {12} n$ | From the definition of excess kurtosis, we have:
:$\gamma_2 = \expect {\paren {\dfrac {X - \mu} \sigma}^4} - 3$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
By Expectation of Chi-Squared Distribution:
:$\mu = n$
By Variance of Chi-Squared Distribution:
:$\sigma = \sqrt {2 n}$
... | Let $n$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $X \sim \chi^2_n$ where $\chi^2_n$ is the [[Definition:Chi-Squared Distribution|chi-squared distribution]] with $n$ degrees of freedom.
Then the [[Definition:Excess Kurtosis|excess kurtosis]] $\gamma_2$ of $X$ is given by:
:$\gam... | From the definition of [[Definition:Excess Kurtosis|excess kurtosis]], we have:
:$\gamma_2 = \expect {\paren {\dfrac {X - \mu} \sigma}^4} - 3$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
By [[Expectation of Chi-S... | Excess Kurtosis of Chi-Squared Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Chi-Squared_Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Chi-Squared_Distribution | [
"Kurtosis",
"Chi-Squared Distribution"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Chi-Squared Distribution",
"Definition:Excess Kurtosis"
] | [
"Definition:Excess Kurtosis",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Chi-Squared Distribution",
"Variance of Chi-Squared Distribution",
"Skewness of Chi-Squared Distribution",
"Raw Moment of Chi-Squared Distribution",
"Binomial Theorem/Examples/4th Power of Sum",
... |
proofwiki-16014 | Raw Moment of Chi-Squared Distribution | Let $n$ and $m$ be strictly positive integers.
Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.
Then the $m$th raw moment $\expect {X^m}$ of $X$ is given by:
:$\ds \expect {X^m} = \prod_{k \mathop = 0}^{m - 1} \paren {n + 2 k}$ | From the definition of the chi-squared distribution, $X$ has probability density function:
:$\ds \map {f_X} x = \dfrac 1 {2^{n / 2} \map \Gamma {n / 2} } x^{\paren {n / 2} - 1} e^{- x / 2}$
From the definition of the expected value of a continuous random variable:
:$\ds \expect {X^m} = \int_0^\infty x^m \map {f_X} x \... | Let $n$ and $m$ be [[Definition:Strictly Positive Integer|strictly positive integers]].
Let $X \sim \chi^2_n$ where $\chi^2_n$ is the [[Definition:Chi-Squared Distribution|chi-squared distribution]] with $n$ degrees of freedom.
Then the $m$th [[Definition:Raw Moment|raw moment]] $\expect {X^m}$ of $X$ is given by: ... | From the definition of the [[Definition:Chi-Squared Distribution|chi-squared distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\ds \map {f_X} x = \dfrac 1 {2^{n / 2} \map \Gamma {n / 2} } x^{\paren {n / 2} - 1} e^{- x / 2}$
From the definition of the [[Definition:Expe... | Raw Moment of Chi-Squared Distribution | https://proofwiki.org/wiki/Raw_Moment_of_Chi-Squared_Distribution | https://proofwiki.org/wiki/Raw_Moment_of_Chi-Squared_Distribution | [
"Raw Moments",
"Chi-Squared Distribution"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Chi-Squared Distribution",
"Definition:Raw Moment"
] | [
"Definition:Chi-Squared Distribution",
"Definition:Probability Density Function",
"Definition:Expectation/Continuous",
"Integration by Substitution",
"Gamma Difference Equation",
"Category:Raw Moments",
"Category:Chi-Squared Distribution"
] |
proofwiki-16015 | Sequence of Square Roots of Natural Numbers is not Cauchy | Let $\sequence {x_n}_{n \mathop \in \N_{>0} }$ be the sequence in $\R$ defined as:
:$x_n = \sqrt n$
Then, despite the fact that from Difference Between Adjacent Square Roots Converges:
:$\size {\sqrt {n + 1} - \sqrt n} \to 0$ as $n \to \infty$
it is not the case that $\sequence {x_n}$ is a Cauchy sequence. | {{AimForCont}} $\sequence {x_n}$ is bounded.
Let $H \in \R$ be an upper bound of $\sequence {x_n}$
By the Axiom of Archimedes:
:$\exists N \in \N: N > H^2$
and so:
:$\exists N \in \N: \sqrt N > H$
But $\sqrt N$ is a term of $\sequence {x_n}$.
So $H$ cannot be an upper bound of $\sequence {x_n}$
Hence by Proof by Contra... | Let $\sequence {x_n}_{n \mathop \in \N_{>0} }$ be the [[Definition:Real Sequence|sequence in $\R$]] defined as:
:$x_n = \sqrt n$
Then, despite the fact that from [[Difference Between Adjacent Square Roots Converges]]:
:$\size {\sqrt {n + 1} - \sqrt n} \to 0$ as $n \to \infty$
it is not the case that $\sequence {x_n}$... | {{AimForCont}} $\sequence {x_n}$ is [[Definition:Bounded Real Sequence|bounded]].
Let $H \in \R$ be an [[Definition:Upper Bound of Real Sequence|upper bound]] of $\sequence {x_n}$
By the [[Axiom of Archimedes]]:
:$\exists N \in \N: N > H^2$
and so:
:$\exists N \in \N: \sqrt N > H$
But $\sqrt N$ is a [[Definition:T... | Sequence of Square Roots of Natural Numbers is not Cauchy | https://proofwiki.org/wiki/Sequence_of_Square_Roots_of_Natural_Numbers_is_not_Cauchy | https://proofwiki.org/wiki/Sequence_of_Square_Roots_of_Natural_Numbers_is_not_Cauchy | [
"Cauchy Sequences"
] | [
"Definition:Real Sequence",
"Difference Between Adjacent Square Roots Converges",
"Definition:Cauchy Sequence/Real Numbers"
] | [
"Definition:Bounded Sequence/Real",
"Definition:Upper Bound of Sequence/Real",
"Axiom of Archimedes",
"Definition:Term of Sequence",
"Definition:Upper Bound of Sequence/Real",
"Proof by Contradiction",
"Definition:Bounded Sequence/Real",
"Cauchy Sequence is Bounded/Real Numbers",
"Rule of Transposit... |
proofwiki-16016 | Excess Kurtosis of Continuous Uniform Distribution | Let $X$ be a continuous random variable which is uniformly distributed on a closed real interval $\closedint a b$.
Then the excess kurtosis $\gamma_2$ of $X$ is equal to $-\dfrac 6 5$. | From the definition of excess kurtosis, we have:
:$\gamma_2 = \expect {\paren {\dfrac {X - \mu} \sigma}^4} - 3$
From the definition of the continuous uniform distribution, $X$ has probability density function:
:$\map {f_X} x = \begin {cases} \dfrac 1 {b - a} & : a \le x \le b \\ 0 & : \text {otherwise} \end {cases}$
... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] which is [[Definition:Continuous Uniform Distribution|uniformly distributed]] on a [[Definition:Closed Real Interval|closed real interval]] $\closedint a b$.
Then the [[Definition:Excess Kurtosis|excess kurtosis]] $\gamma_2$ of $X$ is e... | From the definition of [[Definition:Excess Kurtosis|excess kurtosis]], we have:
:$\gamma_2 = \expect {\paren {\dfrac {X - \mu} \sigma}^4} - 3$
From the definition of the [[Definition:Continuous Uniform Distribution|continuous uniform distribution]], $X$ has [[Definition:Probability Density Function|probability densi... | Excess Kurtosis of Continuous Uniform Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Continuous_Uniform_Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Continuous_Uniform_Distribution | [
"Kurtosis",
"Continuous Uniform Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Uniform Distribution/Continuous",
"Definition:Real Interval/Closed",
"Definition:Excess Kurtosis"
] | [
"Definition:Excess Kurtosis",
"Definition:Uniform Distribution/Continuous",
"Definition:Probability Density Function",
"Definition:Expectation/Continuous",
"Integration by Substitution",
"Primitive of Power",
"Fundamental Theorem of Calculus",
"Variance of Continuous Uniform Distribution",
"Expectat... |
proofwiki-16017 | Square of Standard Normal Random Variable has Chi-Squared Distribution | Let $X \sim \Gaussian 0 1$ where $\Gaussian 0 1$ is the standard normal distribution.
Then $X^2 \sim \chi^2_1$ where $\chi^2_1$ is the chi-square distribution with $1$ degree of freedom. | Let $Y \sim \chi^2_1$.
We aim to show that:
:$\map \Pr {Y < x^2} = \map \Pr {-x < X < x}$
for all real $x \ge 0$.
We have:
{{begin-eqn}}
{{eqn | l = \map \Pr {Y < x^2}
| r = \frac 1 {\sqrt 2 \map \Gamma {1 / 2} } \int_0^{x^2} t^{\paren {1 / 2} - 1} e^{-t / 2} \rd t
| c = {{Defof|Chi-Squared Distribution}... | Let $X \sim \Gaussian 0 1$ where $\Gaussian 0 1$ is the [[Definition:Standard Normal Distribution|standard normal distribution]].
Then $X^2 \sim \chi^2_1$ where $\chi^2_1$ is the [[Definition:Chi-Squared Distribution|chi-square distribution]] with $1$ degree of freedom. | Let $Y \sim \chi^2_1$.
We aim to show that:
:$\map \Pr {Y < x^2} = \map \Pr {-x < X < x}$
for all [[Definition:Real Number|real]] $x \ge 0$.
We have:
{{begin-eqn}}
{{eqn | l = \map \Pr {Y < x^2}
| r = \frac 1 {\sqrt 2 \map \Gamma {1 / 2} } \int_0^{x^2} t^{\paren {1 / 2} - 1} e^{-t / 2} \rd t
| c = {... | Square of Standard Normal Random Variable has Chi-Squared Distribution | https://proofwiki.org/wiki/Square_of_Standard_Normal_Random_Variable_has_Chi-Squared_Distribution | https://proofwiki.org/wiki/Square_of_Standard_Normal_Random_Variable_has_Chi-Squared_Distribution | [
"Normal Distribution",
"Chi-Squared Distribution"
] | [
"Definition:Standard Normal Distribution",
"Definition:Chi-Squared Distribution"
] | [
"Definition:Real Number",
"Gamma Function of One Half",
"Integration by Substitution",
"Definite Integral of Even Function",
"Category:Normal Distribution",
"Category:Chi-Squared Distribution"
] |
proofwiki-16018 | Sum of Chi-Squared Random Variables | Let $n_1, n_2, \ldots, n_k$ be strictly positive integers which sum to $N$.
Let $X_i \sim {\chi^2}_{n_i}$ for $1 \le i \le k$, where ${\chi^2}_{n_i}$ is the chi-squared distribution with $n_i$ degrees of freedom.
Then:
:$\ds X = \sum_{i \mathop = 1}^k X_i \sim {\chi^2}_N$ | Let $Y \sim {\chi^2}_N$.
By Moment Generating Function of Chi-Squared Distribution, the moment generating function of $X_i$ is given by:
:$\map {M_{X_i} } t = \paren {1 - 2 t}^{-n_i / 2}$
Similarly, the moment generating function of $Y$ is given by:
:$\map {M_Y} t = \paren {1 - 2 t}^{-N / 2}$
By Moment Generating Fu... | Let $n_1, n_2, \ldots, n_k$ be [[Definition:Strictly Positive Integer|strictly positive integers]] which sum to $N$.
Let $X_i \sim {\chi^2}_{n_i}$ for $1 \le i \le k$, where ${\chi^2}_{n_i}$ is the [[Definition:Chi-Squared Distribution|chi-squared distribution]] with $n_i$ degrees of freedom.
Then:
:$\ds X = \sum... | Let $Y \sim {\chi^2}_N$.
By [[Moment Generating Function of Chi-Squared Distribution]], the [[Definition:Moment Generating Function|moment generating function]] of $X_i$ is given by:
:$\map {M_{X_i} } t = \paren {1 - 2 t}^{-n_i / 2}$
Similarly, the [[Definition:Moment Generating Function|moment generating function... | Sum of Chi-Squared Random Variables | https://proofwiki.org/wiki/Sum_of_Chi-Squared_Random_Variables | https://proofwiki.org/wiki/Sum_of_Chi-Squared_Random_Variables | [
"Chi-Squared Distribution"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Chi-Squared Distribution"
] | [
"Moment Generating Function of Chi-Squared Distribution",
"Definition:Moment Generating Function",
"Definition:Moment Generating Function",
"Moment Generating Function of Linear Combination of Independent Random Variables",
"Definition:Moment Generating Function",
"Moment Generating Function is Unique",
... |
proofwiki-16019 | Moment Generating Function of Chi-Squared Distribution | Let $n$ be a strictly positive integer.
Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.
Then the moment generating function of $X$, $M_X$, is given by:
:$\map {M_X} t = \begin{cases} \paren {1 - 2 t}^{-n / 2} & : t < \dfrac 1 2 \\ \text{does not exist} & : t \ge \d... | From the definition of the chi-squared distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac 1 {2^{n / 2} \map \Gamma {n / 2} } x^{\paren {n / 2} - 1} e^{- x / 2}$
From the definition of a moment generating function:
:$\ds \map {M_X} t = \expect {e^{t X} } = \int_0^\infty e^{t x} \map {f_X} x \rd ... | Let $n$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $X \sim \chi^2_n$ where $\chi^2_n$ is the [[Definition:Chi-Squared Distribution|chi-squared distribution]] with $n$ degrees of freedom.
Then the [[Definition:Moment Generating Function|moment generating function]] of $X$, $M_X$, is... | From the definition of the [[Definition:Chi-Squared Distribution|chi-squared distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac 1 {2^{n / 2} \map \Gamma {n / 2} } x^{\paren {n / 2} - 1} e^{- x / 2}$
From the definition of a [[Definition:Moment Gen... | Moment Generating Function of Chi-Squared Distribution | https://proofwiki.org/wiki/Moment_Generating_Function_of_Chi-Squared_Distribution | https://proofwiki.org/wiki/Moment_Generating_Function_of_Chi-Squared_Distribution | [
"Moment Generating Functions",
"Chi-Squared Distribution"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Chi-Squared Distribution",
"Definition:Moment Generating Function"
] | [
"Definition:Chi-Squared Distribution",
"Definition:Probability Density Function",
"Definition:Moment Generating Function",
"Exponential Tends to Zero and Infinity",
"Exponential of Zero",
"Integral to Infinity of Reciprocal of Power of x",
"Integration by Substitution"
] |
proofwiki-16020 | Moment Generating Function of Linear Combination of Independent Random Variables | Let $X_1, X_2, \ldots, X_n$ be independent random variables.
Let $k_1, k_2, \ldots, k_n$ be real numbers.
Let:
:$\ds X = \sum_{i \mathop = 1}^n k_i X_i$
Let $M_{X_i}$ be the moment generating function of $X_i$ for $1 \le i \le n$.
Then:
:$\ds \map {M_X} t = \prod_{i \mathop = 1}^n \map {M_{X_i}} {k_i t}$
for all $t$... | {{begin-eqn}}
{{eqn | l = \map {M_X} t
| r = \expect {\map \exp {t X} }
| c = {{Defof|Moment Generating Function}}
}}
{{eqn | r = \expect {\map \exp {t \sum_{i \mathop = 1}^n k_i X_i} }
}}
{{eqn | r = \expect {\prod_{i \mathop = 1}^n \map \exp {t k_i X_i} }
| c = Exponential of Sum
}}
{{eqn | r = \prod_{i \mathop = ... | Let $X_1, X_2, \ldots, X_n$ be [[Definition:Independent Random Variables|independent random variables]].
Let $k_1, k_2, \ldots, k_n$ be [[Definition:Real Number|real numbers]].
Let:
:$\ds X = \sum_{i \mathop = 1}^n k_i X_i$
Let $M_{X_i}$ be the [[Definition:Moment Generating Function|moment generating function]] ... | {{begin-eqn}}
{{eqn | l = \map {M_X} t
| r = \expect {\map \exp {t X} }
| c = {{Defof|Moment Generating Function}}
}}
{{eqn | r = \expect {\map \exp {t \sum_{i \mathop = 1}^n k_i X_i} }
}}
{{eqn | r = \expect {\prod_{i \mathop = 1}^n \map \exp {t k_i X_i} }
| c = [[Exponential of Sum]]
}}
{{eqn | r = \prod_{i \matho... | Moment Generating Function of Linear Combination of Independent Random Variables | https://proofwiki.org/wiki/Moment_Generating_Function_of_Linear_Combination_of_Independent_Random_Variables | https://proofwiki.org/wiki/Moment_Generating_Function_of_Linear_Combination_of_Independent_Random_Variables | [
"Moment Generating Functions"
] | [
"Definition:Independent Random Variables",
"Definition:Real Number",
"Definition:Moment Generating Function"
] | [
"Exponential of Sum",
"Condition for Independence from Product of Expectations",
"Category:Moment Generating Functions"
] |
proofwiki-16021 | Sum of Squares of Standard Normal Random Variables has Chi-Squared Distribution | Let $X_1, X_2, \ldots, X_n$ be independent random variables.
Let $X_i \sim \Gaussian 0 1$ for $1 \le i \le n$ where $\Gaussian 0 1$ is the standard normal distribution.
Then:
:$\ds \sum_{i \mathop = 1}^n X^2_i \sim \chi^2_n$
where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom. | By Square of Standard Normal Random Variable has Chi-Squared Distribution, we have:
:$X^2_i \sim \chi^2_1$
for $1 \le i \le n$.
So, by Sum of Chi-Squared Random Variables, we have:
:$\ds \sum_{i \mathop = 1}^n X^2_i \sim \chi^2_{1 + 1 + 1 \ldots} = \chi^2_n$
{{qed}}
Category:Chi-Squared Distribution
ngbp12fmopqe1mf... | Let $X_1, X_2, \ldots, X_n$ be [[Definition:Independent Random Variables|independent random variables]].
Let $X_i \sim \Gaussian 0 1$ for $1 \le i \le n$ where $\Gaussian 0 1$ is the [[Definition:Standard Normal Distribution|standard normal distribution]].
Then:
:$\ds \sum_{i \mathop = 1}^n X^2_i \sim \chi^2_n$
w... | By [[Square of Standard Normal Random Variable has Chi-Squared Distribution]], we have:
:$X^2_i \sim \chi^2_1$
for $1 \le i \le n$.
So, by [[Sum of Chi-Squared Random Variables]], we have:
:$\ds \sum_{i \mathop = 1}^n X^2_i \sim \chi^2_{1 + 1 + 1 \ldots} = \chi^2_n$
{{qed}}
[[Category:Chi-Squared Distribution]... | Sum of Squares of Standard Normal Random Variables has Chi-Squared Distribution | https://proofwiki.org/wiki/Sum_of_Squares_of_Standard_Normal_Random_Variables_has_Chi-Squared_Distribution | https://proofwiki.org/wiki/Sum_of_Squares_of_Standard_Normal_Random_Variables_has_Chi-Squared_Distribution | [
"Chi-Squared Distribution"
] | [
"Definition:Independent Random Variables",
"Definition:Standard Normal Distribution",
"Definition:Chi-Squared Distribution"
] | [
"Square of Standard Normal Random Variable has Chi-Squared Distribution",
"Sum of Chi-Squared Random Variables",
"Category:Chi-Squared Distribution"
] |
proofwiki-16022 | Variance of Random Sample from Normal Distribution has Chi-Squared Distribution | Let $X_1, X_2, \ldots, X_n$ form a random sample of size $n$ from the normal distribution $\Gaussian \mu {\sigma^2}$ for some $\mu \in \R, \sigma \in \R_{>0}$.
Let:
:$\ds \bar X = \frac 1 n \sum_{i \mathop = 1}^n X_i$
and:
:$\ds s^2 = \frac 1 {n - 1} \sum_{i \mathop = 1}^n \paren {X_i - \bar X}^2$
Then:
:$\dfrac {\pa... | {{ProofWanted}}
Category:Normal Distribution
Category:Chi-Squared Distribution
qnd4tnqldx0ch0su52rzvm28sx9mh8u | Let $X_1, X_2, \ldots, X_n$ form a [[Definition:Random Sample (Probability Theory)|random sample]] of size $n$ from the [[Definition:Normal Distribution|normal distribution]] $\Gaussian \mu {\sigma^2}$ for some $\mu \in \R, \sigma \in \R_{>0}$.
Let:
:$\ds \bar X = \frac 1 n \sum_{i \mathop = 1}^n X_i$
and:
:$\ds s... | {{ProofWanted}}
[[Category:Normal Distribution]]
[[Category:Chi-Squared Distribution]]
qnd4tnqldx0ch0su52rzvm28sx9mh8u | Variance of Random Sample from Normal Distribution has Chi-Squared Distribution | https://proofwiki.org/wiki/Variance_of_Random_Sample_from_Normal_Distribution_has_Chi-Squared_Distribution | https://proofwiki.org/wiki/Variance_of_Random_Sample_from_Normal_Distribution_has_Chi-Squared_Distribution | [
"Normal Distribution",
"Chi-Squared Distribution"
] | [
"Definition:Random Sample (Probability Theory)",
"Definition:Normal Distribution",
"Definition:Chi-Squared Distribution"
] | [
"Category:Normal Distribution",
"Category:Chi-Squared Distribution"
] |
proofwiki-16023 | Expectation of Snedecor's F-Distribution | Let $n, m$ be strictly positive integers.
Let $X \sim F_{n, m}$ where $F_{n, m}$ is Snedecor's $F$-distribution with $\tuple {n, m}$ degrees of freedom.
Then the expectation of $X$ is given by:
:$\expect X = \dfrac m {m - 2}$
for $m > 2$, and does not exist otherwise. | Let $Y$ and $Z$ be independent random variables.
Let $Y \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.
Let $Z \sim \chi^2_m$ where $\chi^2_m$ is the chi-squared distribution with $m$ degrees of freedom.
Then, from Quotient of Independent Random Variables with $\chi^2$ Dist... | Let $n, m$ be [[Definition:Strictly Positive Integer|strictly positive integers]].
Let $X \sim F_{n, m}$ where $F_{n, m}$ is [[Definition:Snedecor's F-Distribution|Snedecor's $F$-distribution]] with $\tuple {n, m}$ degrees of freedom.
Then the [[Definition:Expectation|expectation]] of $X$ is given by:
:$\expect X ... | Let $Y$ and $Z$ be [[Definition:Independent Random Variables|independent random variables]].
Let $Y \sim \chi^2_n$ where $\chi^2_n$ is the [[Definition:Chi-Squared Distribution|chi-squared distribution]] with $n$ degrees of freedom.
Let $Z \sim \chi^2_m$ where $\chi^2_m$ is the [[Definition:Chi-Squared Distribution|... | Expectation of Snedecor's F-Distribution | https://proofwiki.org/wiki/Expectation_of_Snedecor's_F-Distribution | https://proofwiki.org/wiki/Expectation_of_Snedecor's_F-Distribution | [
"Snedecor's F-Distribution",
"Expectation"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Snedecor's F-Distribution",
"Definition:Expectation"
] | [
"Definition:Independent Random Variables",
"Definition:Chi-Squared Distribution",
"Definition:Chi-Squared Distribution",
"Quotient of Independent Random Variables with Chi-Squared Distribution Divided by Degrees of Freedom has Snedecor's F-Distribution",
"Definition:Probability Density Function",
"Definit... |
proofwiki-16024 | Variance of Snedecor's F-Distribution | Let $n, m$ be strictly positive integers.
Let $X \sim F_{n, m}$ where $F_{n, m}$ is Snedecor's $F$-distribution with $\tuple {n, m}$ degrees of freedom.
Then the variance of $X$ is given by:
:$\var X = \dfrac {2 m^2 \paren {m + n - 2} } {n \paren {m - 4} \paren {m - 2}^2}$
for $m > 4$, and does not exist otherwise. | Since $m > 4 > 2$, we have by Expectation of Snedecor's F-Distribution:
:$\expect X = \dfrac m {m - 2}$
We now aim to compute $\expect {X^2}$ with a view to apply Variance as Expectation of Square minus Square of Expectation.
Let $Y$ and $Z$ be independent random variables.
Let $Y \sim \chi^2_n$ where $\chi^2_n$ is the... | Let $n, m$ be [[Definition:Strictly Positive Integer|strictly positive integers]].
Let $X \sim F_{n, m}$ where $F_{n, m}$ is [[Definition:Snedecor's F-Distribution|Snedecor's $F$-distribution]] with $\tuple {n, m}$ degrees of freedom.
Then the [[Definition:Variance|variance]] of $X$ is given by:
:$\var X = \dfrac... | Since $m > 4 > 2$, we have by [[Expectation of Snedecor's F-Distribution]]:
:$\expect X = \dfrac m {m - 2}$
We now aim to compute $\expect {X^2}$ with a view to apply [[Variance as Expectation of Square minus Square of Expectation]].
Let $Y$ and $Z$ be [[Definition:Independent Random Variables|independent random var... | Variance of Snedecor's F-Distribution | https://proofwiki.org/wiki/Variance_of_Snedecor's_F-Distribution | https://proofwiki.org/wiki/Variance_of_Snedecor's_F-Distribution | [
"Variance",
"Snedecor's F-Distribution"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Snedecor's F-Distribution",
"Definition:Variance"
] | [
"Expectation of Snedecor's F-Distribution",
"Variance as Expectation of Square minus Square of Expectation",
"Definition:Independent Random Variables",
"Definition:Chi-Squared Distribution",
"Definition:Chi-Squared Distribution",
"Definition:Probability Density Function",
"Definition:Joint Probability D... |
proofwiki-16025 | Differential Equations for Shortest Path on 3d Sphere/Cartesian Coordinates | Let $M$ be a $3$-dimensional Euclidean space.
Let $S$ be a sphere embedded in $M$.
Let $\gamma$ be a curve on $S$.
Let the chosen coordinate system be Cartesian.
Let $\gamma$ begin at $\paren {x_0, y_0, z_0}$ and terminate at $\paren {x_1, y_1, z_1}$.
Let $\map y x$, $\map z x$ be real functions.
Let $\gamma$ connectin... | In $3$-dimensional Euclidean space the length of the curve is:
:$\ds \int_{x_0}^{x_1} \sqrt {1 + y'^2 + z'^2} \rd x$
The sphere satisfies the following equation:
{{begin-eqn}}
{{eqn | l = \map g {x, y, z}
| r = x^2 + y^2 + z^2 - a^2
}}
{{eqn | r = 0
}}
{{end-eqn}}
Consider its partial derivatives {{WRT|Differenti... | Let $M$ be a [[Definition:Dimension of Vector Space|$3$-dimensional]] [[Definition:Real Euclidean Space|Euclidean space]].
Let $S$ be a [[Definition:Sphere|sphere]] embedded in $M$.
Let $\gamma$ be a [[Definition:Curve|curve]] on $S$.
Let the chosen [[Definition:Coordinate System|coordinate system]] be [[Definition:... | In [[Definition:Dimension of Vector Space|$3$-dimensional]] [[Definition:Real Euclidean Space|Euclidean space]] the [[Definition:Arc Length|length]] of the [[Definition:Curve|curve]] is:
:$\ds \int_{x_0}^{x_1} \sqrt {1 + y'^2 + z'^2} \rd x$
The [[Definition:Sphere|sphere]] satisfies the following equation:
{{begin-e... | Differential Equations for Shortest Path on 3d Sphere/Cartesian Coordinates | https://proofwiki.org/wiki/Differential_Equations_for_Shortest_Path_on_3d_Sphere/Cartesian_Coordinates | https://proofwiki.org/wiki/Differential_Equations_for_Shortest_Path_on_3d_Sphere/Cartesian_Coordinates | [
"Calculus of Variations"
] | [
"Definition:Dimension of Vector Space",
"Definition:Euclidean Space/Real",
"Definition:Sphere",
"Definition:Line/Curve",
"Definition:Coordinate System",
"Definition:Cartesian Coordinate System",
"Definition:Real Function",
"Definition:Directed Smooth Curve/Endpoints",
"Definition:Minimum Value of Fu... | [
"Definition:Dimension of Vector Space",
"Definition:Euclidean Space/Real",
"Definition:Arc Length",
"Definition:Line/Curve",
"Definition:Sphere",
"Definition:Partial Derivative/Real Analysis",
"Definition:Sphere",
"Definition:Point",
"Simplest Variational Problem with Subsidiary Conditions for Curve... |
proofwiki-16026 | Skewness of Snedecor's F-Distribution | Let $n, m$ be strictly positive integers.
Let $X \sim F_{n, m}$ where $F_{n, m}$ is Snedecor's $F$-distribution with $\tuple {n, m}$ degrees of freedom.
Then the skewness $\gamma_1$ of $X$ is given by:
:$\gamma_1 = \dfrac {2 \paren {m + 2 n - 2} } {m - 6} \sqrt {\dfrac {2 \paren {m - 4} } {n \paren {m + n - 2} } }$
f... | {{ProofWanted}}
Category:Snedecor's F-Distribution
Category:Skewness
5izl4gd0tmfwpnlsqnpt49clukg58ay | Let $n, m$ be [[Definition:Strictly Positive Integer|strictly positive integers]].
Let $X \sim F_{n, m}$ where $F_{n, m}$ is [[Definition:Snedecor's F-Distribution|Snedecor's $F$-distribution]] with $\tuple {n, m}$ degrees of freedom.
Then the [[Definition:Skewness|skewness]] $\gamma_1$ of $X$ is given by:
:$\gam... | {{ProofWanted}}
[[Category:Snedecor's F-Distribution]]
[[Category:Skewness]]
5izl4gd0tmfwpnlsqnpt49clukg58ay | Skewness of Snedecor's F-Distribution | https://proofwiki.org/wiki/Skewness_of_Snedecor's_F-Distribution | https://proofwiki.org/wiki/Skewness_of_Snedecor's_F-Distribution | [
"Snedecor's F-Distribution",
"Skewness"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Snedecor's F-Distribution",
"Definition:Skewness"
] | [
"Category:Snedecor's F-Distribution",
"Category:Skewness"
] |
proofwiki-16027 | Excess Kurtosis of Snedecor's F-Distribution | Let $n, m$ be strictly positive integers.
Let $X \sim F_{n, m}$ where $F_{n, m}$ is Snedecor's F-distribution with $\tuple {n, m}$ degrees of freedom.
Then the excess kurtosis $\gamma_2$ of $X$ is given by:
:$\gamma_2 = \dfrac {12 \paren {5 m n^2 - 22 n^2 + 5 m^2 n - 32 m n + 44 n + m^3 - 8 m^2 + 20 m - 16} } {n \par... | {{ProofWanted}}
Category:Kurtosis
Category:Snedecor's F-Distribution
43hnxj7j0bsok0d5pguqmn6gmsrrgbz | Let $n, m$ be [[Definition:Strictly Positive Integer|strictly positive integers]].
Let $X \sim F_{n, m}$ where $F_{n, m}$ is [[Definition:Snedecor's F-Distribution|Snedecor's F-distribution]] with $\tuple {n, m}$ degrees of freedom.
Then the [[Definition:Excess Kurtosis|excess kurtosis]] $\gamma_2$ of $X$ is given ... | {{ProofWanted}}
[[Category:Kurtosis]]
[[Category:Snedecor's F-Distribution]]
43hnxj7j0bsok0d5pguqmn6gmsrrgbz | Excess Kurtosis of Snedecor's F-Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Snedecor's_F-Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Snedecor's_F-Distribution | [
"Kurtosis",
"Snedecor's F-Distribution"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Snedecor's F-Distribution",
"Definition:Excess Kurtosis"
] | [
"Category:Kurtosis",
"Category:Snedecor's F-Distribution"
] |
proofwiki-16028 | Expectation of Chi Distribution | Let $n$ be a strictly positive integer.
Let $X \sim \chi_n$ where $\chi_n$ is the chi distribution with $n$ degrees of freedom.
Then the expectation of $X$ is given by:
:$\expect X = \sqrt 2 \dfrac {\map \Gamma {\paren {n + 1} / 2} } {\map \Gamma {n / 2} }$
where $\Gamma$ is the gamma function. | From the definition of the chi distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac 1 {2^{\paren {n / 2} - 1} \map \Gamma {n / 2} } x^{n - 1} e^{- x^2 / 2}$
From the definition of the expected value of a continuous random variable:
:$\ds \expect X = \int_0^\infty x \map {f_X} x \rd x$
So:
{{begin... | Let $n$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $X \sim \chi_n$ where $\chi_n$ is the [[Definition:Chi Distribution|chi distribution]] with $n$ degrees of freedom.
Then the [[Definition:Expectation|expectation]] of $X$ is given by:
:$\expect X = \sqrt 2 \dfrac {\map \Gamma {\... | From the definition of the [[Definition:Chi Distribution|chi distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac 1 {2^{\paren {n / 2} - 1} \map \Gamma {n / 2} } x^{n - 1} e^{- x^2 / 2}$
From the definition of the [[Definition:Expectation of Continu... | Expectation of Chi Distribution | https://proofwiki.org/wiki/Expectation_of_Chi_Distribution | https://proofwiki.org/wiki/Expectation_of_Chi_Distribution | [
"Expectation",
"Chi Distribution"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Chi Distribution",
"Definition:Expectation",
"Definition:Gamma Function"
] | [
"Definition:Chi Distribution",
"Definition:Probability Density Function",
"Definition:Expectation/Continuous",
"Integration by Substitution",
"Category:Expectation",
"Category:Chi Distribution"
] |
proofwiki-16029 | Mean of Random Sample from Chi-Squared Distribution has Gamma Distribution | Let $n$ be a strictly positive integer.
Let $X_1, X_2, \ldots, X_k$ form a random sample of size $k$ from the chi-squared distribution with $n$ degrees of freedom.
Then:
:$\ds \overline X = \frac 1 k \sum_{i \mathop = 1}^k X_i \sim \map \Gamma {\frac {n k} 2, \frac k 2}$
where $\map \Gamma {\dfrac {n k} 2, \dfrac k ... | By Sum of Chi-Squared Random Variables, we have:
:$\ds \sum_{i \mathop = 1}^k X_i \sim \chi^2_{n k}$
By Multiple of Chi-Squared Random Variable has Gamma Distribution, we then have:
:$\ds \frac 1 k \sum_{i \mathop = 1}^k X_i \sim \map \Gamma {\frac {n k} 2, \frac 1 {\frac 2 k} } = \map \Gamma {\frac {n k} 2, \frac k ... | Let $n$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $X_1, X_2, \ldots, X_k$ form a [[Definition:Random Sample (Probability Theory)|random sample]] of size $k$ from the [[Definition:Chi-Squared Distribution|chi-squared distribution with $n$ degrees of freedom]].
Then:
:$\ds \overl... | By [[Sum of Chi-Squared Random Variables]], we have:
:$\ds \sum_{i \mathop = 1}^k X_i \sim \chi^2_{n k}$
By [[Multiple of Chi-Squared Random Variable has Gamma Distribution]], we then have:
:$\ds \frac 1 k \sum_{i \mathop = 1}^k X_i \sim \map \Gamma {\frac {n k} 2, \frac 1 {\frac 2 k} } = \map \Gamma {\frac {n k} ... | Mean of Random Sample from Chi-Squared Distribution has Gamma Distribution | https://proofwiki.org/wiki/Mean_of_Random_Sample_from_Chi-Squared_Distribution_has_Gamma_Distribution | https://proofwiki.org/wiki/Mean_of_Random_Sample_from_Chi-Squared_Distribution_has_Gamma_Distribution | [
"Chi-Squared Distribution",
"Gamma Distribution"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Random Sample (Probability Theory)",
"Definition:Chi-Squared Distribution",
"Definition:Gamma Distribution"
] | [
"Sum of Chi-Squared Random Variables",
"Multiple of Chi-Squared Random Variable has Gamma Distribution",
"Category:Chi-Squared Distribution",
"Category:Gamma Distribution"
] |
proofwiki-16030 | Mode of Normal Distribution | Let $\mu$ be a real number.
Let $\sigma$ be a strictly positive real number.
Let $X \sim \Gaussian \mu {\sigma^2}$ where $\Gaussian \mu {\sigma^2}$ is the normal distribution with parameters $\mu$ and $\sigma^2$.
Then the mode $M$ of $X$ is equal to $\mu$. | By the definition of the normal distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac 1 {\sigma \sqrt {2 \pi} } \, \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$
By the definition of a mode, $f_X$ attains its global maximum at $M$.
From Exponential is Strictly Increasing, we have that ... | Let $\mu$ be a [[Definition:Real Number|real number]].
Let $\sigma$ be a [[Definition:Strictly Positive/Real Number|strictly positive real number]].
Let $X \sim \Gaussian \mu {\sigma^2}$ where $\Gaussian \mu {\sigma^2}$ is the [[Definition:Normal Distribution|normal distribution]] with parameters $\mu$ and $\sigma^... | By the definition of the [[Definition:Normal Distribution|normal distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac 1 {\sigma \sqrt {2 \pi} } \, \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$
By the definition of a [[Definition:Mode of C... | Mode of Normal Distribution | https://proofwiki.org/wiki/Mode_of_Normal_Distribution | https://proofwiki.org/wiki/Mode_of_Normal_Distribution | [
"Modes",
"Normal Distribution"
] | [
"Definition:Real Number",
"Definition:Strictly Positive/Real Number",
"Definition:Normal Distribution",
"Definition:Mode of Continuous Random Variable"
] | [
"Definition:Normal Distribution",
"Definition:Probability Density Function",
"Definition:Mode of Continuous Random Variable",
"Exponential is Strictly Increasing",
"Category:Modes",
"Category:Normal Distribution"
] |
proofwiki-16031 | Equivalence of Definitions of Separated Sets | Let $T = \struct {S, \tau}$ be a topological space.
Let $A, B \subseteq S$.
{{TFAE|def = Separated Sets}} | === Definition 1 implies Definition 2 ===
{{:Equivalence of Definitions of Separated Sets/Definition 1 implies Definition 2}}{{qed|lemma}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A, B \subseteq S$.
{{TFAE|def = Separated Sets}} | === [[Equivalence of Definitions of Separated Sets/Definition 1 implies Definition 2|Definition 1 implies Definition 2]] ===
{{:Equivalence of Definitions of Separated Sets/Definition 1 implies Definition 2}}{{qed|lemma}} | Equivalence of Definitions of Separated Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Separated_Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Separated_Sets | [
"Separated Sets"
] | [
"Definition:Topological Space"
] | [
"Equivalence of Definitions of Separated Sets/Definition 1 implies Definition 2"
] |
proofwiki-16032 | Equivalence of Definitions of Separated Sets/Definition 1 implies Definition 2 | Let $A, B \subseteq S$ satisfy:
:$A^- \cap B = A \cap B^- = \O$
where $A^-$ denotes the closure of $A$ in $T$, and $\O$ denotes the empty set. | From Topological Closure is Closed, $B^-$ is closed in $T$.
Let $U = S \setminus B^-$ be the relative complement of $B^-$.
By the definition of a closed set, $U$ is open in $T$.
From Empty Intersection iff Subset of Relative Complement:
:$A \subseteq S \setminus B^- = U$
From Relative Complement of Relative Complement... | Let $A, B \subseteq S$ satisfy:
:$A^- \cap B = A \cap B^- = \O$
where $A^-$ denotes the [[Definition:Closure (Topology)|closure]] of $A$ in $T$, and $\O$ denotes the [[Definition:Empty Set|empty set]]. | From [[Topological Closure is Closed]], $B^-$ is [[Definition:Closed Set (Topology)|closed]] in $T$.
Let $U = S \setminus B^-$ be the [[Definition:Relative Complement|relative complement]] of $B^-$.
By the definition of a [[Definition:Closed Set (Topology)|closed set]], $U$ is [[Definition:Open Set (Topology)|open]]... | Equivalence of Definitions of Separated Sets/Definition 1 implies Definition 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Separated_Sets/Definition_1_implies_Definition_2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Separated_Sets/Definition_1_implies_Definition_2 | [
"Separated Sets"
] | [
"Definition:Closure (Topology)",
"Definition:Empty Set"
] | [
"Topological Closure is Closed",
"Definition:Closed Set/Topology",
"Definition:Relative Complement",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Empty Intersection iff Subset of Relative Complement",
"Relative Complement of Relative Complement",
"Definition:Closure (Topology)",
... |
proofwiki-16033 | Equivalence of Definitions of Separated Sets/Definition 2 implies Definition 1 | Let $A, B \subseteq S$.
Let $U, V \in \tau$ satisfy:
:$A \subset U$ and $U \cap B = \O$
:$B \subset V$ and $V \cap A = \O$ | Let $U \in \tau$ be an arbitrary open set of $T$ such that $A \subseteq U$ and $U \cap B = \O$.
From Empty Intersection iff Subset of Relative Complement:
:$B \subseteq S \setminus U$
By the definition of a closed set, the relative complement $S \setminus U$ of $U$ is closed in $T$.
From Set Closure is Smallest Closed ... | Let $A, B \subseteq S$.
Let $U, V \in \tau$ satisfy:
:$A \subset U$ and $U \cap B = \O$
:$B \subset V$ and $V \cap A = \O$ | Let $U \in \tau$ be an arbitrary [[Definition:Open Set (Topology)|open set]] of $T$ such that $A \subseteq U$ and $U \cap B = \O$.
From [[Empty Intersection iff Subset of Relative Complement]]:
:$B \subseteq S \setminus U$
By the definition of a [[Definition:Closed Set (Topology)|closed set]], the [[Definition:Relati... | Equivalence of Definitions of Separated Sets/Definition 2 implies Definition 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Separated_Sets/Definition_2_implies_Definition_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Separated_Sets/Definition_2_implies_Definition_1 | [
"Separated Sets"
] | [] | [
"Definition:Open Set/Topology",
"Empty Intersection iff Subset of Relative Complement",
"Definition:Closed Set/Topology",
"Definition:Relative Complement",
"Definition:Closed Set/Topology",
"Set Closure is Smallest Closed Set/Topology",
"Empty Intersection iff Subset of Relative Complement",
"Intersec... |
proofwiki-16034 | Expectation of Student's t-Distribution | Let $k$ be a strictly positive integer.
Let $X \sim t_k$ where $t_k$ is the $t$-distribution with $k$ degrees of freedom.
Then the expectation of $X$ is equal to $0$ for $k > 1$, and does not exist otherwise. | From the definition of the Student's t-Distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac {\map \Gamma {\frac {k + 1} 2} } {\sqrt {\pi k} \map \Gamma {\frac k 2} } \paren {1 + \dfrac {x^2} k}^{-\frac {k + 1} 2}$
with $k$ degrees of freedom for some $k \in \R_{>0}$.
From the definition of the ex... | Let $k$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $X \sim t_k$ where $t_k$ is the [[Definition:Student's t-Distribution|$t$-distribution]] with $k$ degrees of freedom.
Then the [[Definition:Expectation|expectation]] of $X$ is equal to $0$ for $k > 1$, and does not exist otherwise. | From the definition of the [[Definition:Student's t-Distribution|Student's t-Distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac {\map \Gamma {\frac {k + 1} 2} } {\sqrt {\pi k} \map \Gamma {\frac k 2} } \paren {1 + \dfrac {x^2} k}^{-\frac {k + 1} 2}... | Expectation of Student's t-Distribution | https://proofwiki.org/wiki/Expectation_of_Student's_t-Distribution | https://proofwiki.org/wiki/Expectation_of_Student's_t-Distribution | [
"Expectation",
"Student's t-Distribution"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Student's t-Distribution",
"Definition:Expectation"
] | [
"Definition:Student's t-Distribution",
"Definition:Probability Density Function",
"Definition:Expectation/Continuous",
"Sum of Integrals on Adjacent Intervals for Continuous Functions",
"Integration by Substitution/Definite Integral",
"Reversal of Limits of Definite Integral",
"Primitive of Power",
"F... |
proofwiki-16035 | Variance of Student's t-Distribution | Let $k$ be a strictly positive integer.
Let $X \sim t_k$ where $t_k$ is the $t$-distribution with $k$ degrees of freedom.
Then the variance of $X$ is given by:
:$\var X = \dfrac k {k - 2}$
for $k > 2$, and does not exist otherwise. | By Expectation of Student's t-Distribution, we have that for $k > 1$:
:$\expect X = 0$
From Square of Random Variable with t-Distribution has Snedecor's F-Distribution, we have:
:$\expect {X^2} = \expect Y$
with $Y \sim F_{1, k}$, where $F_{1, k}$ is Snedecor's $F$-distribution with $\tuple {1, k}$ degrees of freedom... | Let $k$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $X \sim t_k$ where $t_k$ is the [[Definition:Student's t-Distribution|$t$-distribution]] with $k$ degrees of freedom.
Then the [[Definition:Variance|variance]] of $X$ is given by:
:$\var X = \dfrac k {k - 2}$
for $k > 2$, and do... | By [[Expectation of Student's t-Distribution]], we have that for $k > 1$:
:$\expect X = 0$
From [[Square of Random Variable with t-Distribution has Snedecor's F-Distribution]], we have:
:$\expect {X^2} = \expect Y$
with $Y \sim F_{1, k}$, where $F_{1, k}$ is [[Definition:Snedecor's F-Distribution|Snedecor's $F$-d... | Variance of Student's t-Distribution/Proof 1 | https://proofwiki.org/wiki/Variance_of_Student's_t-Distribution | https://proofwiki.org/wiki/Variance_of_Student's_t-Distribution/Proof_1 | [
"Variance",
"Student's t-Distribution"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Student's t-Distribution",
"Definition:Variance"
] | [
"Expectation of Student's t-Distribution",
"Square of Random Variable with t-Distribution has Snedecor's F-Distribution",
"Definition:Snedecor's F-Distribution",
"Expectation of Snedecor's F-Distribution",
"Variance as Expectation of Square minus Square of Expectation"
] |
proofwiki-16036 | Excess Kurtosis of Student's t-Distribution | Let $k$ be a strictly positive integer.
Let $X \sim t_k$ where $t_k$ is the $t$-distribution with $k$ degrees of freedom.
Then the excess kurtosis $\gamma_2$ of $X$ is given by:
:$\gamma_2 = \dfrac 6 {k - 4}$
for $k > 4$, and does not exist otherwise. | {{ProofWanted}}
Category:Kurtosis
Category:Student's t-Distribution
fh5e6p2nrsroc8oef1stmqrg3tfq1a5 | Let $k$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $X \sim t_k$ where $t_k$ is the [[Definition:Student's t-Distribution|$t$-distribution]] with $k$ degrees of freedom.
Then the [[Definition:Excess Kurtosis|excess kurtosis]] $\gamma_2$ of $X$ is given by:
:$\gamma_2 = \dfrac 6 {k... | {{ProofWanted}}
[[Category:Kurtosis]]
[[Category:Student's t-Distribution]]
fh5e6p2nrsroc8oef1stmqrg3tfq1a5 | Excess Kurtosis of Student's t-Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Student's_t-Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Student's_t-Distribution | [
"Kurtosis",
"Student's t-Distribution"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Student's t-Distribution",
"Definition:Excess Kurtosis"
] | [
"Category:Kurtosis",
"Category:Student's t-Distribution"
] |
proofwiki-16037 | Connected Set in Subspace | Let $T = \struct {S, \tau}$ be a topological space.
Let $A \subseteq B \subseteq S$.
Let $T_B = \struct {B, \tau_B}$ be the topological space where $\tau_B$ is the subspace topology on $B$.
Then
:$A$ is connected in $T_B$ {{iff}} $A$ is connected in $T$. | Let $\tau_A$ be the subspace topology on $A$ induced by $\tau$.
Let $\tau'_A$ be the subspace topology on $A$ induced by $\tau_B$.
By the definition of a connected set, $A$ is connected in $T$ {{iff}} $\struct {A, \tau_A}$ is a connected topological space.
Similarly, $A$ is connected in $T_B$ {{iff}} $\struct {A, \tau'... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A \subseteq B \subseteq S$.
Let $T_B = \struct {B, \tau_B}$ be the [[Definition:Topological Space|topological space]] where $\tau_B$ is the [[Definition:Subspace Topology|subspace topology]] on $B$.
Then
:$A$ is [[Definition:C... | Let $\tau_A$ be the [[Definition:Subspace Topology|subspace topology]] on $A$ induced by $\tau$.
Let $\tau'_A$ be the [[Definition:Subspace Topology|subspace topology]] on $A$ induced by $\tau_B$.
By the definition of a [[Definition:Connected Set (Topology)|connected set]], $A$ is [[Definition:Connected Set (Topology... | Connected Set in Subspace | https://proofwiki.org/wiki/Connected_Set_in_Subspace | https://proofwiki.org/wiki/Connected_Set_in_Subspace | [
"Connected Topological Spaces"
] | [
"Definition:Topological Space",
"Definition:Topological Space",
"Definition:Topological Subspace",
"Definition:Connected Set (Topology)",
"Definition:Connected Set (Topology)"
] | [
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Definition:Connected Set (Topology)",
"Definition:Connected Set (Topology)",
"Definition:Connected Topological Space",
"Definition:Connected Set (Topology)",
"Definition:Connected Topological Space",
"Subspace of Subspace is Subspa... |
proofwiki-16038 | Raw Moment of Erlang Distribution | Let $n, k$ be strictly positive integers.
Let $\lambda$ be a strictly positive real number.
Let $X$ have a continuous random variable with an Erlang distribution with parameters $k$ and $\lambda$.
Then the $n$th raw moment of $X$ is given by:
:$\ds \expect {X^n} = \frac 1 {\lambda^n} \prod_{m \mathop = 0}^{n - 1} \pa... | From the definition of the Erlang distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac {\lambda^k x^{k - 1} e^{- \lambda x} } {\map \Gamma k}$
From the definition of the expected value of a continuous random variable:
:$\ds \expect {X^n} = \int_0^\infty x^n \map {f_X} x \rd x$
So:
{{begin-eqn}}
... | Let $n, k$ be [[Definition:Strictly Positive Integer|strictly positive integers]].
Let $\lambda$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $X$ have a [[Definition:Continuous Random Variable|continuous random variable]] with an [[Definition:Erlang Distribution|Erlang distri... | From the definition of the [[Definition:Erlang Distribution|Erlang distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac {\lambda^k x^{k - 1} e^{- \lambda x} } {\map \Gamma k}$
From the definition of the [[Definition:Expectation of Continuous Random ... | Raw Moment of Erlang Distribution | https://proofwiki.org/wiki/Raw_Moment_of_Erlang_Distribution | https://proofwiki.org/wiki/Raw_Moment_of_Erlang_Distribution | [
"Raw Moments",
"Erlang Distribution"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Strictly Positive/Real Number",
"Definition:Random Variable/Continuous",
"Definition:Erlang Distribution",
"Definition:Raw Moment"
] | [
"Definition:Erlang Distribution",
"Definition:Probability Density Function",
"Definition:Expectation/Continuous",
"Integration by Substitution",
"Gamma Difference Equation",
"Category:Raw Moments",
"Category:Erlang Distribution"
] |
proofwiki-16039 | Expectation of Erlang Distribution | Let $k$ be a strictly positive integer.
Let $\lambda$ be a strictly positive real number.
Let $X$ be a continuous random variable with an Erlang distribution with parameters $k$ and $\lambda$.
Then the expectation of $X$ is given by:
:$\expect X = \dfrac k \lambda$ | {{begin-eqn}}
{{eqn | l = \expect X
| r = \frac 1 {\lambda^1} \prod_{m \mathop = 0}^0 \paren {k + m}
| c = Raw Moment of Erlang Distribution
}}
{{eqn | r = \frac k \lambda
}}
{{end-eqn}}
{{qed}}
Category:Expectation
Category:Erlang Distribution
2mvhbomtjhf2h95radm04h4kd0kzyb2 | Let $k$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $\lambda$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with an [[Definition:Erlang Distribution|Erlang distributi... | {{begin-eqn}}
{{eqn | l = \expect X
| r = \frac 1 {\lambda^1} \prod_{m \mathop = 0}^0 \paren {k + m}
| c = [[Raw Moment of Erlang Distribution]]
}}
{{eqn | r = \frac k \lambda
}}
{{end-eqn}}
{{qed}}
[[Category:Expectation]]
[[Category:Erlang Distribution]]
2mvhbomtjhf2h95radm04h4kd0kzyb2 | Expectation of Erlang Distribution | https://proofwiki.org/wiki/Expectation_of_Erlang_Distribution | https://proofwiki.org/wiki/Expectation_of_Erlang_Distribution | [
"Expectation",
"Erlang Distribution"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Strictly Positive/Real Number",
"Definition:Random Variable/Continuous",
"Definition:Erlang Distribution",
"Definition:Expectation"
] | [
"Raw Moment of Erlang Distribution",
"Category:Expectation",
"Category:Erlang Distribution"
] |
proofwiki-16040 | Variance of Erlang Distribution | Let $k$ be a strictly positive integer.
Let $\lambda$ be a strictly positive real number.
Let $X$ be a continuous random variable with an Erlang distribution with parameters $k$ and $\lambda$.
Then the variance of $X$ is given by:
:$\var X = \dfrac k {\lambda^2}$ | By Variance as Expectation of Square minus Square of Expectation, we have:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
By Expectation of Erlang Distribution, we have:
:$\expect X = \dfrac k \lambda$
We also have:
{{begin-eqn}}
{{eqn | l = \expect {X^2}
| r = \frac 1 {\lambda^2} \prod_{m \mathop = 0}^1 \paren {... | Let $k$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $\lambda$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with an [[Definition:Erlang Distribution|Erlang distributi... | By [[Variance as Expectation of Square minus Square of Expectation]], we have:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
By [[Expectation of Erlang Distribution]], we have:
:$\expect X = \dfrac k \lambda$
We also have:
{{begin-eqn}}
{{eqn | l = \expect {X^2}
| r = \frac 1 {\lambda^2} \prod_{m \mathop = ... | Variance of Erlang Distribution | https://proofwiki.org/wiki/Variance_of_Erlang_Distribution | https://proofwiki.org/wiki/Variance_of_Erlang_Distribution | [
"Variance",
"Erlang Distribution"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Strictly Positive/Real Number",
"Definition:Random Variable/Continuous",
"Definition:Erlang Distribution",
"Definition:Variance"
] | [
"Variance as Expectation of Square minus Square of Expectation",
"Expectation of Erlang Distribution",
"Category:Variance",
"Category:Erlang Distribution"
] |
proofwiki-16041 | Roots of Complex Number/Examples/z^4 - 81 = 0 | The roots of the polynomial:
:$z^4 - 81$
are:
:$\set {3, 3 i, -3, -3 i}$ | From Factorisation of $z^n - a$:
:$z^4 - a = \ds \prod_{k \mathop = 0}^3 \paren {z - \alpha^k b}$
where:
:$\alpha$ is a primitive complex $4$th root of unity
:$b$ is any complex number such that $b^4 = a$.
Here we can take $b = 3$, as $81 = 3^4$.
Thus:
:$z = \set {3 \exp \dfrac {k i \pi} 2}$
{{begin-eqn}}
{{eqn | n = k... | The [[Definition:Root of Polynomial|roots]] of the [[Definition:Polynomial over Complex Numbers|polynomial]]:
:$z^4 - 81$
are:
:$\set {3, 3 i, -3, -3 i}$ | From [[Factorisation of z^n-a|Factorisation of $z^n - a$]]:
:$z^4 - a = \ds \prod_{k \mathop = 0}^3 \paren {z - \alpha^k b}$
where:
:$\alpha$ is a [[Definition:Primitive Complex Root of Unity|primitive complex $4$th root of unity]]
:$b$ is any [[Definition:Complex Number|complex number]] such that $b^4 = a$.
Here we ... | Roots of Complex Number/Examples/z^4 - 81 = 0 | https://proofwiki.org/wiki/Roots_of_Complex_Number/Examples/z^4_-_81_=_0 | https://proofwiki.org/wiki/Roots_of_Complex_Number/Examples/z^4_-_81_=_0 | [
"Examples of Complex Roots"
] | [
"Definition:Root of Polynomial",
"Definition:Polynomial/Complex Numbers"
] | [
"Factorisation of z^n-a",
"Definition:Root of Unity/Complex/Primitive",
"Definition:Complex Number"
] |
proofwiki-16042 | Covariance as Expectation of Product minus Product of Expectations | Let $X$ and $Y$ be random variables.
Let the expectations of $X$ and $Y$ exist and be finite.
Then the covariance of $X$ and $Y$ is given by:
:$\cov {X, Y} = \expect {X Y} - \expect X \expect Y$ | {{begin-eqn}}
{{eqn | l = \cov {X, Y}
| r = \expect {\paren {X - \expect X} \paren {Y - \expect Y} }
| c = {{Defof|Covariance}}
}}
{{eqn | r = \expect {X Y - X \expect Y - Y \expect X + \expect X \expect Y}
}}
{{eqn | r = \expect {X Y} - \expect Y \expect X - \expect Y \expect X + \expect X \expect Y
... | Let $X$ and $Y$ be [[Definition:Random Variable|random variables]].
Let the [[Definition:Expectation|expectations]] of $X$ and $Y$ exist and be finite.
Then the [[Definition:Covariance|covariance]] of $X$ and $Y$ is given by:
:$\cov {X, Y} = \expect {X Y} - \expect X \expect Y$ | {{begin-eqn}}
{{eqn | l = \cov {X, Y}
| r = \expect {\paren {X - \expect X} \paren {Y - \expect Y} }
| c = {{Defof|Covariance}}
}}
{{eqn | r = \expect {X Y - X \expect Y - Y \expect X + \expect X \expect Y}
}}
{{eqn | r = \expect {X Y} - \expect Y \expect X - \expect Y \expect X + \expect X \expect Y
... | Covariance as Expectation of Product minus Product of Expectations | https://proofwiki.org/wiki/Covariance_as_Expectation_of_Product_minus_Product_of_Expectations | https://proofwiki.org/wiki/Covariance_as_Expectation_of_Product_minus_Product_of_Expectations | [
"Covariance"
] | [
"Definition:Random Variable",
"Definition:Expectation",
"Definition:Covariance"
] | [
"Expectation is Linear",
"Expectation of Constant"
] |
proofwiki-16043 | Covariance of Independent Random Variables is Zero | Let $X$ and $Y$ be independent random variables.
Let the expectations of $X$ and $Y$ exist and be finite.
Then the covariance of $X$ and $Y$ is $0$. | {{begin-eqn}}
{{eqn | l = \cov {X, Y}
| r = \expect {X Y} - \expect X \expect Y
| c = Covariance as Expectation of Product minus Product of Expectations
}}
{{eqn | r = \expect X \expect Y - \expect X \expect Y
| c = Condition for Independence from Product of Expectations
}}
{{eqn | r = 0
}}
{{end-eqn}... | Let $X$ and $Y$ be [[Definition:Independent Random Variables|independent random variables]].
Let the [[Definition:Expectation|expectations]] of $X$ and $Y$ exist and be finite.
Then the [[Definition:Covariance|covariance]] of $X$ and $Y$ is $0$. | {{begin-eqn}}
{{eqn | l = \cov {X, Y}
| r = \expect {X Y} - \expect X \expect Y
| c = [[Covariance as Expectation of Product minus Product of Expectations]]
}}
{{eqn | r = \expect X \expect Y - \expect X \expect Y
| c = [[Condition for Independence from Product of Expectations]]
}}
{{eqn | r = 0
}}
{{... | Covariance of Independent Random Variables is Zero | https://proofwiki.org/wiki/Covariance_of_Independent_Random_Variables_is_Zero | https://proofwiki.org/wiki/Covariance_of_Independent_Random_Variables_is_Zero | [
"Covariance"
] | [
"Definition:Independent Random Variables",
"Definition:Expectation",
"Definition:Covariance"
] | [
"Covariance as Expectation of Product minus Product of Expectations",
"Condition for Independence from Product of Expectations"
] |
proofwiki-16044 | Variance of Linear Combination of Random Variables | Let $X$ and $Y$ be random variables.
Let the variances of $X$ and $Y$ be finite.
Let $a$ and $b$ be real numbers.
Then the variance of $a X + b Y$ is given by:
:$\var {a X + b Y} = a^2 \, \var X + b^2 \, \var Y + 2 a b \, \cov {X, Y}$
where $\cov {X, Y}$ is the covariance of $X$ and $Y$. | {{begin-eqn}}
{{eqn | l = \var {a X + b Y}
| r = \expect {\paren {a X + b Y - \expect {a X + b Y} }^2}
| c = {{Defof|Variance}}
}}
{{eqn | r = \expect {\paren {a X + b Y - a \, \expect X - b \, \expect Y}^2}
| c = Expectation is Linear
}}
{{eqn | r = \expect {\paren {a \paren {X - \expect X} + b \pare... | Let $X$ and $Y$ be [[Definition:Random Variable|random variables]].
Let the [[Definition:Variance|variances]] of $X$ and $Y$ be finite.
Let $a$ and $b$ be [[Definition:Real Number|real numbers]].
Then the [[Definition:Variance|variance]] of $a X + b Y$ is given by:
:$\var {a X + b Y} = a^2 \, \var X + b^2 \, \... | {{begin-eqn}}
{{eqn | l = \var {a X + b Y}
| r = \expect {\paren {a X + b Y - \expect {a X + b Y} }^2}
| c = {{Defof|Variance}}
}}
{{eqn | r = \expect {\paren {a X + b Y - a \, \expect X - b \, \expect Y}^2}
| c = [[Expectation is Linear]]
}}
{{eqn | r = \expect {\paren {a \paren {X - \expect X} + b \... | Variance of Linear Combination of Random Variables | https://proofwiki.org/wiki/Variance_of_Linear_Combination_of_Random_Variables | https://proofwiki.org/wiki/Variance_of_Linear_Combination_of_Random_Variables | [
"Variance"
] | [
"Definition:Random Variable",
"Definition:Variance",
"Definition:Real Number",
"Definition:Variance",
"Definition:Covariance"
] | [
"Expectation is Linear",
"Expectation is Linear",
"Category:Variance"
] |
proofwiki-16045 | Square of Expectation of Product is Less Than or Equal to Product of Expectation of Squares | Let $X$ and $Y$ be random variables.
Let the expectation of $X Y$, $\expect {X Y}$, exist and be finite.
Then:
:$\paren {\expect {X Y} }^2 \le \expect {X^2} \expect {Y^2}$ | Note that:
:$\map \Pr {Y^2 \ge 0} = 1$
so we have by Expectation of Non-Negative Random Variable is Non-Negative:
:$\expect {Y^2} \ge 0$
First, take $\expect {Y^2} > 0$.
Let $Z$ be a random variable with:
:$Z = X - Y \dfrac {\expect {X Y} } {\expect {Y^2} }$
Note that we have:
:$\map \Pr {Z^2 \ge 0} = 1$
so again ... | Let $X$ and $Y$ be [[Definition:Random Variable|random variables]].
Let the [[Definition:Expectation|expectation]] of $X Y$, $\expect {X Y}$, exist and be finite.
Then:
:$\paren {\expect {X Y} }^2 \le \expect {X^2} \expect {Y^2}$ | Note that:
:$\map \Pr {Y^2 \ge 0} = 1$
so we have by [[Expectation of Non-Negative Random Variable is Non-Negative]]:
:$\expect {Y^2} \ge 0$
First, take $\expect {Y^2} > 0$.
Let $Z$ be a [[Definition:Random Variable|random variable]] with:
:$Z = X - Y \dfrac {\expect {X Y} } {\expect {Y^2} }$
Note that we ha... | Square of Expectation of Product is Less Than or Equal to Product of Expectation of Squares | https://proofwiki.org/wiki/Square_of_Expectation_of_Product_is_Less_Than_or_Equal_to_Product_of_Expectation_of_Squares | https://proofwiki.org/wiki/Square_of_Expectation_of_Product_is_Less_Than_or_Equal_to_Product_of_Expectation_of_Squares | [
"Expectation"
] | [
"Definition:Random Variable",
"Definition:Expectation"
] | [
"Expectation of Non-Negative Random Variable is Non-Negative",
"Definition:Random Variable",
"Expectation of Non-Negative Random Variable is Non-Negative",
"Square of Sum",
"Expectation is Linear",
"Condition for Expectation of Non-Negative Random Variable to be Zero",
"Definition:Random Variable",
"E... |
proofwiki-16046 | Square of Covariance is Less Than or Equal to Product of Variances | Let $X$ and $Y$ be random variables.
Let the variances of $X$ and $Y$ exist and be finite.
Then:
:$\paren {\cov {X, Y} }^2 \le \var X \, \var Y$
where $\cov {X, Y}$ denotes the covariance of $X$ and $Y$. | We have, by the definition of variance, that both:
:$\expect {\paren {X - \expect X}^2}$
and:
:$\expect {\paren {Y - \expect Y}^2}$
exist and are finite.
Therefore:
{{begin-eqn}}
{{eqn | l = \paren {\cov {X, Y} }^2
| r = \paren {\expect {\paren {X - \expect X} \paren {Y - \expect Y} } }^2
| c = {{Defof|C... | Let $X$ and $Y$ be [[Definition:Random Variable|random variables]].
Let the [[Definition:Variance|variances]] of $X$ and $Y$ exist and be finite.
Then:
:$\paren {\cov {X, Y} }^2 \le \var X \, \var Y$
where $\cov {X, Y}$ denotes the [[Definition:Covariance|covariance]] of $X$ and $Y$. | We have, by the definition of [[Definition:Variance|variance]], that both:
:$\expect {\paren {X - \expect X}^2}$
and:
:$\expect {\paren {Y - \expect Y}^2}$
exist and are finite.
Therefore:
{{begin-eqn}}
{{eqn | l = \paren {\cov {X, Y} }^2
| r = \paren {\expect {\paren {X - \expect X} \paren {Y - \expect... | Square of Covariance is Less Than or Equal to Product of Variances | https://proofwiki.org/wiki/Square_of_Covariance_is_Less_Than_or_Equal_to_Product_of_Variances | https://proofwiki.org/wiki/Square_of_Covariance_is_Less_Than_or_Equal_to_Product_of_Variances | [
"Covariance",
"Variance"
] | [
"Definition:Random Variable",
"Definition:Variance",
"Definition:Covariance"
] | [
"Definition:Variance",
"Square of Expectation of Product is Less Than or Equal to Product of Expectation of Squares"
] |
proofwiki-16047 | Absolute Value of Pearson Correlation Coefficient is Less Than or Equal to 1 | Let $X$ and $Y$ be random variables.
Let the variances of $X$ and $Y$ exist and be finite.
Then:
:$\size {\map \rho {X, Y} } \le 1$
where $\map \rho {X, Y}$ denotes the Pearson correlation coefficient of $X$ and $Y$. | {{begin-eqn}}
{{eqn | l = \paren {\map \rho {X, Y} }^2
| r = \paren {\frac {\map {\operatorname {Cov} } {X, Y} } {\sqrt {\var X \var Y} } }^2
| c = {{Defof|Pearson Correlation Coefficient}}
}}
{{eqn | r = \frac {\paren {\map {\operatorname {Cov} } {X, Y} }^2} {\var X \var Y}
}}
{{eqn | o = \le
| r = 1... | Let $X$ and $Y$ be [[Definition:Random Variable|random variables]].
Let the [[Definition:Variance|variances]] of $X$ and $Y$ exist and be finite.
Then:
:$\size {\map \rho {X, Y} } \le 1$
where $\map \rho {X, Y}$ denotes the [[Definition:Pearson Correlation Coefficient|Pearson correlation coefficient]] of $X$ an... | {{begin-eqn}}
{{eqn | l = \paren {\map \rho {X, Y} }^2
| r = \paren {\frac {\map {\operatorname {Cov} } {X, Y} } {\sqrt {\var X \var Y} } }^2
| c = {{Defof|Pearson Correlation Coefficient}}
}}
{{eqn | r = \frac {\paren {\map {\operatorname {Cov} } {X, Y} }^2} {\var X \var Y}
}}
{{eqn | o = \le
| r = 1... | Absolute Value of Pearson Correlation Coefficient is Less Than or Equal to 1 | https://proofwiki.org/wiki/Absolute_Value_of_Pearson_Correlation_Coefficient_is_Less_Than_or_Equal_to_1 | https://proofwiki.org/wiki/Absolute_Value_of_Pearson_Correlation_Coefficient_is_Less_Than_or_Equal_to_1 | [
"Pearson Correlation Coefficient"
] | [
"Definition:Random Variable",
"Definition:Variance",
"Definition:Pearson Correlation Coefficient"
] | [
"Square of Covariance is Less Than or Equal to Product of Variances",
"Cauchy's Inequality"
] |
proofwiki-16048 | Equivalence of Definitions of Locally Connected Space/Definition 1 implies Definition 2 | Let each point of $T$ have a local basis consisting entirely of connected sets in $T$. | From Local Basis for Open Sets Implies Neighborhood Basis of Open Sets, it follows directly that:
:each point of $T$ has a neighborhood basis consisting entirely of connected sets in $T$. | Let each point of $T$ have a [[Definition:Local Basis|local basis]] consisting entirely of [[Definition:Connected Set (Topology)|connected sets]] in $T$. | From [[Equivalence of Definitions of Local Basis/Local Basis for Open Sets Implies Neighborhood Basis of Open Sets|Local Basis for Open Sets Implies Neighborhood Basis of Open Sets]], it follows directly that:
:each point of $T$ has a [[Definition:Neighborhood Basis|neighborhood basis]] consisting entirely of [[Defini... | Equivalence of Definitions of Locally Connected Space/Definition 1 implies Definition 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Connected_Space/Definition_1_implies_Definition_2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Connected_Space/Definition_1_implies_Definition_2 | [
"Equivalence of Definitions of Locally Connected Space"
] | [
"Definition:Local Basis",
"Definition:Connected Set (Topology)"
] | [
"Equivalence of Definitions of Local Basis/Local Basis for Open Sets Implies Neighborhood Basis of Open Sets",
"Definition:Neighborhood Basis",
"Definition:Connected Set (Topology)"
] |
proofwiki-16049 | Equivalence of Definitions of Locally Connected Space/Definition 2 implies Definition 1 | Let $T$ be weakly locally connected at each point of $T$.
That is, each point of $T$ has a neighborhood basis consisting of connected sets of $T$. | Let $x \in S$ and $x \in U \in \tau$.
Let $\BB_x = \set {W \in \tau : x \in W, W \text{ is connected in } T}$.
By definition of local basis, we have to show that there exists a connected open set $V \in \BB_x$ with $x \in V \subset U$.
Let $V = \map {\operatorname{Comp}_x} U$ denote the component of $x$ in $U$.
From Op... | Let $T$ be [[Definition:Weakly Locally Connected at Point|weakly locally connected at]] each point of $T$.
That is, each point of $T$ has a [[Definition:Neighborhood Basis|neighborhood basis]] consisting of [[Definition:Connected Set (Topology)|connected sets]] of $T$. | Let $x \in S$ and $x \in U \in \tau$.
Let $\BB_x = \set {W \in \tau : x \in W, W \text{ is connected in } T}$.
By definition of [[Definition:Local Basis|local basis]], we have to show that there exists a [[Definition:Connected Set (Topology)|connected]] [[Definition:Open Set (Topology)|open set]] $V \in \BB_x$ with ... | Equivalence of Definitions of Locally Connected Space/Definition 2 implies Definition 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Connected_Space/Definition_2_implies_Definition_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Connected_Space/Definition_2_implies_Definition_1 | [
"Equivalence of Definitions of Locally Connected Space"
] | [
"Definition:Weakly Locally Connected at Point",
"Definition:Neighborhood Basis",
"Definition:Connected Set (Topology)"
] | [
"Definition:Local Basis",
"Definition:Connected Set (Topology)",
"Definition:Open Set/Topology",
"Definition:Component (Topology)",
"Open Set in Open Subspace",
"Definition:Open Set/Topology",
"Set is Open iff Neighborhood of all its Points",
"Definition:Neighborhood (Topology)/Point",
"Definition:C... |
proofwiki-16050 | Equivalence of Definitions of Locally Connected Space/Definition 1 implies Definition 3 | Let each point $x$ of $T$ have a local basis $\DD_x$ consisting entirely of connected sets in $T$. | By definition of local basis, each of these connected sets in $\DD_x$ is open in $T$.
Consider the set $\ds \DD = \bigcup_{x \mathop \in S} \DD_x$.
From Union of Local Bases is Basis, $\DD$ is a basis for the topology $\tau$.
Since each $\DD_x$ consists entirely of connected sets, $\DD$ also consists entirely of connec... | Let each point $x$ of $T$ have a [[Definition:Local Basis|local basis]] $\DD_x$ consisting entirely of [[Definition:Connected Set (Topology)|connected sets]] in $T$. | By definition of [[Definition:Local Basis|local basis]], each of these [[Definition:Connected Set (Topology)|connected sets]] in $\DD_x$ is [[Definition:Open Set (Topology)|open]] in $T$.
Consider the set $\ds \DD = \bigcup_{x \mathop \in S} \DD_x$.
From [[Union of Local Bases is Basis]], $\DD$ is a [[Definition:Basi... | Equivalence of Definitions of Locally Connected Space/Definition 1 implies Definition 3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Connected_Space/Definition_1_implies_Definition_3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Connected_Space/Definition_1_implies_Definition_3 | [
"Equivalence of Definitions of Locally Connected Space"
] | [
"Definition:Local Basis",
"Definition:Connected Set (Topology)"
] | [
"Definition:Local Basis",
"Definition:Connected Set (Topology)",
"Definition:Open Set/Topology",
"Union of Local Bases is Basis",
"Definition:Basis (Topology)",
"Definition:Topology",
"Definition:Connected Set (Topology)",
"Definition:Connected Set (Topology)"
] |
proofwiki-16051 | Equivalence of Definitions of Locally Connected Space/Definition 3 implies Definition 1 | Let $T$ have a basis $\BB$ consisting of connected sets in $T$. | For each $x \in S$ we define:
:$\BB_x = \set {B \in \BB: x \in B}$
From Basis induces Local Basis, $\BB_x$ is a local basis.
As each element of $\BB_x$ is also an element of $\BB$, it follows that $\BB_x$ is also formed of connected sets.
Thus, for each point $x \in S$, there is a local basis which consists entirely of... | Let $T$ have a [[Definition:Analytic Basis|basis]] $\BB$ consisting of [[Definition:Connected Set (Topology)|connected sets]] in $T$. | For each $x \in S$ we define:
:$\BB_x = \set {B \in \BB: x \in B}$
From [[Basis induces Local Basis]], $\BB_x$ is a [[Definition:Local Basis|local basis]].
As each [[Definition:Element|element]] of $\BB_x$ is also an [[Definition:Element|element]] of $\BB$, it follows that $\BB_x$ is also formed of [[Definition:Conne... | Equivalence of Definitions of Locally Connected Space/Definition 3 implies Definition 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Connected_Space/Definition_3_implies_Definition_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Connected_Space/Definition_3_implies_Definition_1 | [
"Equivalence of Definitions of Locally Connected Space"
] | [
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Connected Set (Topology)"
] | [
"Basis induces Local Basis",
"Definition:Local Basis",
"Definition:Element",
"Definition:Element",
"Definition:Connected Set (Topology)",
"Definition:Local Basis",
"Definition:Connected Set (Topology)"
] |
proofwiki-16052 | Equivalence of Definitions of Locally Connected Space/Definition 3 implies Definition 4 | Let $T$ have a basis consisting of connected sets in $T$. | Let $U$ be an open subset of $T$.
From Open Set is Union of Elements of Basis, $U$ is a union of open connected sets in $T$.
From Open Set in Open Subspace and Connected Set in Subspace, $U$ is a union of open connected sets in $U$.
From Components are Open iff Union of Open Connected Sets, the components of $U$ are op... | Let $T$ have a [[Definition:Analytic Basis|basis]] consisting of [[Definition:Connected Set (Topology)|connected sets]] in $T$. | Let $U$ be an [[Definition:Open Set (Topology)|open subset]] of $T$.
From [[Open Set is Union of Elements of Basis]], $U$ is a [[Definition:Set Union|union]] of [[Definition:Open Set (Topology)|open]] [[Definition:Connected Set (Topology)|connected sets]] in $T$.
From [[Open Set in Open Subspace]] and [[Connected Set... | Equivalence of Definitions of Locally Connected Space/Definition 3 implies Definition 4 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Connected_Space/Definition_3_implies_Definition_4 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Connected_Space/Definition_3_implies_Definition_4 | [
"Equivalence of Definitions of Locally Connected Space"
] | [
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Connected Set (Topology)"
] | [
"Definition:Open Set/Topology",
"Open Set is Union of Elements of Basis",
"Definition:Set Union",
"Definition:Open Set/Topology",
"Definition:Connected Set (Topology)",
"Open Set in Open Subspace",
"Connected Set in Subspace",
"Definition:Set Union",
"Definition:Open Set/Topology",
"Definition:Con... |
proofwiki-16053 | Equivalence of Definitions of Locally Connected Space/Definition 4 implies Definition 3 | Let the components of the open sets of $T$ are also open in $T$. | Let $\BB = \set {U \in \tau : U \text{ is connected in } T}$.
Let $U$ be open in $T$.
By assumption, the components of $U$ are open in $T$.
From Connected Set in Subspace, the components of $U$ are connected in $T$.
By the definition of the components of a topological space, $U$ is the union of its components.
Hence $U... | Let the [[Definition:Component (Topology)|components]] of the [[Definition:Open Set (Topology)|open sets]] of $T$ are also [[Definition:Open Set (Topology)|open]] in $T$. | Let $\BB = \set {U \in \tau : U \text{ is connected in } T}$.
Let $U$ be open in $T$.
By assumption, the [[Definition:Component (Topology)|components]] of $U$ are [[Definition:Open Set (Topology)|open]] in $T$.
From [[Connected Set in Subspace]], the [[Definition:Component (Topology)|components]] of $U$ are [[Defini... | Equivalence of Definitions of Locally Connected Space/Definition 4 implies Definition 3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Connected_Space/Definition_4_implies_Definition_3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Connected_Space/Definition_4_implies_Definition_3 | [
"Equivalence of Definitions of Locally Connected Space"
] | [
"Definition:Component (Topology)",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology"
] | [
"Definition:Component (Topology)",
"Definition:Open Set/Topology",
"Connected Set in Subspace",
"Definition:Component (Topology)",
"Definition:Connected Set (Topology)",
"Definition:Component (Topology)",
"Definition:Topological Space",
"Definition:Set Union",
"Definition:Component (Topology)",
"D... |
proofwiki-16054 | Connected Component is Closed | Let $T = \struct {S, \tau}$ be a topological space.
Then every connected component of $T$ is closed. | Let $H$ be a connected component of $T$.
By the definition of connected component, $H$ is connected.
From Closure of Connected Set is Connected then the closure $H^-$ is connected.
By the definition of the closure, $H \subseteq H^-$.
By the definition of connected component, $H$ is a maximal connected set.
Hence $H = H... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Then every [[Definition:Component (Topology)|connected component]] of $T$ is [[Definition:Closed Set|closed]]. | Let $H$ be a [[Definition:Component (Topology)|connected component]] of $T$.
By the definition of [[Definition:Component (Topology)|connected component]], $H$ is [[Definition:Connected Set (Topology)|connected]].
From [[Closure of Connected Set is Connected]] then the [[Definition:Closure|closure]] $H^-$ is [[Definit... | Connected Component is Closed | https://proofwiki.org/wiki/Connected_Component_is_Closed | https://proofwiki.org/wiki/Connected_Component_is_Closed | [
"Components (Topology)",
"Closed Sets"
] | [
"Definition:Topological Space",
"Definition:Component (Topology)",
"Definition:Closed Set"
] | [
"Definition:Component (Topology)",
"Definition:Component (Topology)",
"Definition:Connected Set (Topology)",
"Closure of Connected Set is Connected",
"Definition:Closure",
"Definition:Connected Set (Topology)",
"Definition:Closure",
"Definition:Component (Topology)",
"Definition:Maximal/Set",
"Def... |
proofwiki-16055 | Constant Mapping is Non-Commutative | Let $S$ be a set whose cardinality is greater than one.
Let $f: S \to S$ and $g: S \to S$ be constant mappings on $S$.
Then:
:$f \circ g \ne g \circ f$
where $\circ$ denotes composition of mappings. | First note that if $S$ is a singleton, then there exists only one constant mapping on $S$.
In such a circumstance, $f = g$ and so $f \circ g \ne g \circ f$.
{{qed|lemma}}
So, let $\card S > 1$.
Then there exist at least $2$ distinct elements $a$ and $b$ of $S$.
Thus, let $f$ and $g$ be defined as:
:$\forall x \in S: \m... | Let $S$ be a [[Definition:Set|set]] whose [[Definition:Cardinality|cardinality]] is greater than one.
Let $f: S \to S$ and $g: S \to S$ be [[Definition:Constant Mapping|constant mappings]] on $S$.
Then:
:$f \circ g \ne g \circ f$
where $\circ$ denotes [[Definition:Composition of Mappings|composition of mappings]]. | First note that if $S$ is a [[Definition:Singleton|singleton]], then there exists only one [[Definition:Constant Mapping|constant mapping]] on $S$.
In such a circumstance, $f = g$ and so $f \circ g \ne g \circ f$.
{{qed|lemma}}
So, let $\card S > 1$.
Then there exist at least $2$ [[Definition:Distinct Elements|dist... | Constant Mapping is Non-Commutative | https://proofwiki.org/wiki/Constant_Mapping_is_Non-Commutative | https://proofwiki.org/wiki/Constant_Mapping_is_Non-Commutative | [
"Constant Mappings"
] | [
"Definition:Set",
"Definition:Cardinality",
"Definition:Constant Mapping",
"Definition:Composition of Mappings"
] | [
"Definition:Singleton",
"Definition:Constant Mapping",
"Definition:Distinct/Plural"
] |
proofwiki-16056 | Quotient Mapping is Injection iff Equality | Let $\RR$ be an equivalence relation on $S$.
Then the quotient mapping $q_\RR: S \to S / \RR$ is an injection {{iff}} $\RR$ is the equality relation. | Let $\eqclass x \RR, \eqclass y \RR \in S / \RR$ | Let $\RR$ be an [[Definition:Equivalence Relation|equivalence relation]] on $S$.
Then the [[Definition:Quotient Mapping|quotient mapping]] $q_\RR: S \to S / \RR$ is an [[Definition:Injection|injection]] {{iff}} $\RR$ is the [[Definition:Equals|equality relation]]. | Let $\eqclass x \RR, \eqclass y \RR \in S / \RR$ | Quotient Mapping is Injection iff Equality | https://proofwiki.org/wiki/Quotient_Mapping_is_Injection_iff_Equality | https://proofwiki.org/wiki/Quotient_Mapping_is_Injection_iff_Equality | [
"Quotient Sets",
"Injections"
] | [
"Definition:Equivalence Relation",
"Definition:Quotient Mapping",
"Definition:Injection",
"Definition:Equals"
] | [] |
proofwiki-16057 | Closure of Subset in Subspace | Let $T = \struct{S, \tau}$ be a topological space.
Let $H \subseteq S$ be an arbitrary subset of $S$.
Let $T_H = \struct {H, \tau_H}$ be the topological subspace on $H$.
Let $A \subseteq H$ be an arbitrary subset of $H$.
Then:
:$\map {\cl_H} A = H \cap \map \cl A$
where:
:$\map {\cl_H} A$ denotes the closure of $A$ in ... | {{begin-eqn}}
{{eqn | l = \map {\cl_H} A
| r = \bigcap \set {K \subseteq H: A \subseteq K, K \text{ is closed in } T_H}
| c = {{Defof|Closure (Topology)}}
}}
{{eqn | r = \bigcap \set {N \cap H: A \subseteq N, N \text{ is closed in } T}
| c = Closed Set in Topological Subspace
}}
{{eqn | r = H \cap \bi... | Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq S$ be an arbitrary [[Definition:Subset|subset]] of $S$.
Let $T_H = \struct {H, \tau_H}$ be the [[Definition:Topological Subspace|topological subspace]] on $H$.
Let $A \subseteq H$ be an arbitrary [[Definition:Subset|... | {{begin-eqn}}
{{eqn | l = \map {\cl_H} A
| r = \bigcap \set {K \subseteq H: A \subseteq K, K \text{ is closed in } T_H}
| c = {{Defof|Closure (Topology)}}
}}
{{eqn | r = \bigcap \set {N \cap H: A \subseteq N, N \text{ is closed in } T}
| c = [[Closed Set in Topological Subspace]]
}}
{{eqn | r = H \cap... | Closure of Subset in Subspace | https://proofwiki.org/wiki/Closure_of_Subset_in_Subspace | https://proofwiki.org/wiki/Closure_of_Subset_in_Subspace | [
"Closure of Subset in Subspace",
"Set Closures",
"Topological Subspaces"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Topological Subspace",
"Definition:Subset",
"Definition:Closure (Topology)",
"Definition:Closure (Topology)"
] | [
"Closed Set in Topological Subspace",
"Set Intersection is Self-Distributive/Families of Sets"
] |
proofwiki-16058 | Equivalence of Definitions of Locally Path-Connected Space/Definition 1 implies Definition 2 | Let each point of $T$ have a local basis consisting entirely of path-connected sets in $T$. | From Local Basis for Open Sets Implies Neighborhood Basis of Open Sets, it follows directly that:
:each point of $T$ has a neighborhood basis consisting entirely of path-connected sets in $T$. | Let each point of $T$ have a [[Definition:Local Basis|local basis]] consisting entirely of [[Definition:Path-Connected Set|path-connected sets]] in $T$. | From [[Equivalence of Definitions of Local Basis/Local Basis for Open Sets Implies Neighborhood Basis of Open Sets|Local Basis for Open Sets Implies Neighborhood Basis of Open Sets]], it follows directly that:
:each point of $T$ has a [[Definition:Neighborhood Basis|neighborhood basis]] consisting entirely of [[Defini... | Equivalence of Definitions of Locally Path-Connected Space/Definition 1 implies Definition 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Path-Connected_Space/Definition_1_implies_Definition_2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Path-Connected_Space/Definition_1_implies_Definition_2 | [
"Equivalence of Definitions of Locally Path-Connected Space"
] | [
"Definition:Local Basis",
"Definition:Path-Connected/Set"
] | [
"Equivalence of Definitions of Local Basis/Local Basis for Open Sets Implies Neighborhood Basis of Open Sets",
"Definition:Neighborhood Basis",
"Definition:Path-Connected/Set"
] |
proofwiki-16059 | Equivalence of Definitions of Locally Path-Connected Space/Definition 2 implies Definition 1 | Let each point of $T$ have a neighborhood basis consisting of path-connected sets in $T$. | Let $x \in S$ and $x \in U \in \tau$.
Let $\BB_x = \set {W \in \tau : x \in W, W \text { is path-connected in } T}$.
By definition of local basis, we have to show that there exists a path-connected open set $V \in \tau$ with $x \in V \subset U$.
Let $V = \map {\operatorname {PC}_x} U$ denote the path component of $x$ i... | Let each point of $T$ have a [[Definition:Neighborhood Basis|neighborhood basis]] consisting of [[Definition:Path-Connected Set|path-connected sets]] in $T$. | Let $x \in S$ and $x \in U \in \tau$.
Let $\BB_x = \set {W \in \tau : x \in W, W \text { is path-connected in } T}$.
By definition of [[Definition:Local Basis|local basis]], we have to show that there exists a [[Definition:Path-Connected Set|path-connected]] [[Definition:Open Set (Topology)|open set]] $V \in \tau$ w... | Equivalence of Definitions of Locally Path-Connected Space/Definition 2 implies Definition 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Path-Connected_Space/Definition_2_implies_Definition_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Path-Connected_Space/Definition_2_implies_Definition_1 | [
"Equivalence of Definitions of Locally Path-Connected Space"
] | [
"Definition:Neighborhood Basis",
"Definition:Path-Connected/Set"
] | [
"Definition:Local Basis",
"Definition:Path-Connected/Set",
"Definition:Open Set/Topology",
"Definition:Path Component",
"Open Set in Open Subspace",
"Definition:Open Set/Topology",
"Set is Open iff Neighborhood of all its Points",
"Definition:Neighborhood (Topology)/Point",
"Definition:Path-Connecte... |
proofwiki-16060 | Equivalence of Definitions of Locally Path-Connected Space/Definition 1 implies Definition 3 | Let each point $x$ of $T$ have a local basis $\DD_x$ consisting entirely of path-connected sets in $T$. | By definition of local basis, each of these path-connected sets in $\DD_x$ is open in $T$.
Consider the set $\ds \DD = \bigcup_{x \mathop \in S} \DD_x$.
From Union of Local Bases is Basis, $\DD$ is a basis for the topology $\tau$.
Since each $\DD_x$ consists entirely of path-connected sets, $\DD$ also consists entirely... | Let each point $x$ of $T$ have a [[Definition:Local Basis|local basis]] $\DD_x$ consisting entirely of [[Definition:Path-Connected Set|path-connected sets]] in $T$. | By definition of [[Definition:Local Basis|local basis]], each of these [[Definition:Path-Connected Set|path-connected sets]] in $\DD_x$ is [[Definition:Open Set (Topology)|open]] in $T$.
Consider the set $\ds \DD = \bigcup_{x \mathop \in S} \DD_x$.
From [[Union of Local Bases is Basis]], $\DD$ is a [[Definition:Basis... | Equivalence of Definitions of Locally Path-Connected Space/Definition 1 implies Definition 3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Path-Connected_Space/Definition_1_implies_Definition_3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Path-Connected_Space/Definition_1_implies_Definition_3 | [
"Equivalence of Definitions of Locally Path-Connected Space"
] | [
"Definition:Local Basis",
"Definition:Path-Connected/Set"
] | [
"Definition:Local Basis",
"Definition:Path-Connected/Set",
"Definition:Open Set/Topology",
"Union of Local Bases is Basis",
"Definition:Basis (Topology)",
"Definition:Topology",
"Definition:Path-Connected/Set",
"Definition:Path-Connected/Set"
] |
proofwiki-16061 | Equivalence of Definitions of Locally Path-Connected Space/Definition 3 implies Definition 1 | Let $T$ have a basis $\BB$ consisting of path-connected sets in $T$. | For each $x \in S$ we define:
:$\BB_x = \set {B \in \BB: x \in B}$
From Basis induces Local Basis, $\BB_x$ is a local basis.
As each element of $\BB_x$ is also an element of $\BB$, it follows that $\BB_x$ is also formed of path-connected sets.
Thus, for each point $x \in S$, there is a local basis which consists entire... | Let $T$ have a [[Definition:Basis (Topology)|basis]] $\BB$ consisting of [[Definition:Path-Connected Set|path-connected sets]] in $T$. | For each $x \in S$ we define:
:$\BB_x = \set {B \in \BB: x \in B}$
From [[Basis induces Local Basis]], $\BB_x$ is a [[Definition:Local Basis|local basis]].
As each [[Definition:Element|element]] of $\BB_x$ is also an [[Definition:Element|element]] of $\BB$, it follows that $\BB_x$ is also formed of [[Definition:Path-... | Equivalence of Definitions of Locally Path-Connected Space/Definition 3 implies Definition 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Path-Connected_Space/Definition_3_implies_Definition_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Path-Connected_Space/Definition_3_implies_Definition_1 | [
"Equivalence of Definitions of Locally Path-Connected Space"
] | [
"Definition:Basis (Topology)",
"Definition:Path-Connected/Set"
] | [
"Basis induces Local Basis",
"Definition:Local Basis",
"Definition:Element",
"Definition:Element",
"Definition:Path-Connected/Set",
"Definition:Local Basis",
"Definition:Path-Connected/Set"
] |
proofwiki-16062 | Equivalence of Definitions of Locally Path-Connected Space/Definition 3 implies Definition 4 | Let $T$ have a basis consisting of path-connected sets in $T$. | Let $U$ be an open subset of $T$.
From Open Set is Union of Elements of Basis, $U$ is a union of open path-connected sets in $T$.
From Open Set in Open Subspace and Path-Connected Set in Subspace, $U$ is a union of open path-connected sets in $U$.
From Path Components are Open iff Union of Open Path-Connected Sets, the... | Let $T$ have a [[Definition:Basis (Topology)|basis]] consisting of [[Definition:Path-Connected Set|path-connected sets]] in $T$. | Let $U$ be an [[Definition:Open Set (Topology)|open subset]] of $T$.
From [[Open Set is Union of Elements of Basis]], $U$ is a [[Definition:Set Union|union]] of [[Definition:Open Set (Topology)|open]] [[Definition:Path-Connected Set|path-connected sets]] in $T$.
From [[Open Set in Open Subspace]] and [[Path-Connected... | Equivalence of Definitions of Locally Path-Connected Space/Definition 3 implies Definition 4 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Path-Connected_Space/Definition_3_implies_Definition_4 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Path-Connected_Space/Definition_3_implies_Definition_4 | [
"Equivalence of Definitions of Locally Path-Connected Space"
] | [
"Definition:Basis (Topology)",
"Definition:Path-Connected/Set"
] | [
"Definition:Open Set/Topology",
"Open Set is Union of Elements of Basis",
"Definition:Set Union",
"Definition:Open Set/Topology",
"Definition:Path-Connected/Set",
"Open Set in Open Subspace",
"Path-Connected Set in Subspace",
"Definition:Set Union",
"Definition:Open Set/Topology",
"Definition:Path... |
proofwiki-16063 | Equivalence of Definitions of Locally Path-Connected Space/Definition 4 implies Definition 3 | Let the path components of open sets of $T$ be also open in $T$. | Let $\BB = \set {U \in \tau : U \text{ is path-connected in } T}$.
Let $U$ be open in $T$.
By assumption, the path components of $U$ are open in $T$.
From Path-Connected Set in Subspace, the path components of $U$ are path-connected in $T$.
By the definition of the path components of a topological space, $U$ is the uni... | Let the [[Definition:Path Component|path components]] of [[Definition:Open Set (Topology)|open sets]] of $T$ be also [[Definition:Open Set (Topology)|open]] in $T$. | Let $\BB = \set {U \in \tau : U \text{ is path-connected in } T}$.
Let $U$ be open in $T$.
By assumption, the [[Definition:Path Component|path components]] of $U$ are [[Definition:Open Set (Topology)|open]] in $T$.
From [[Path-Connected Set in Subspace]], the [[Definition:Path Component|path components]] of $U$ are ... | Equivalence of Definitions of Locally Path-Connected Space/Definition 4 implies Definition 3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Path-Connected_Space/Definition_4_implies_Definition_3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Path-Connected_Space/Definition_4_implies_Definition_3 | [
"Equivalence of Definitions of Locally Path-Connected Space"
] | [
"Definition:Path Component",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology"
] | [
"Definition:Path Component",
"Definition:Open Set/Topology",
"Path-Connected Set in Subspace",
"Definition:Path Component",
"Definition:Path-Connected/Set",
"Definition:Path Component",
"Definition:Topological Space",
"Definition:Set Union",
"Definition:Path Component",
"Definition:Set Union",
"... |
proofwiki-16064 | Binomial Coefficient 2 n Choose n is Divisible by All Primes between n and 2 n | Let $\dbinom {2 n} n$ denote a binomial coefficient.
Then for all prime numbers $p$ such that $n < p < 2 n$:
:$p \divides \dbinom {2 n} n$
where $\divides$ denotes divisibility. | By definition of binomial coefficient:
{{begin-eqn}}
{{eqn | l = \dbinom {2 n} n
| r = \dfrac {\paren {2 n}!} {\paren {n!}^2}
| c = where $n!$ denotes the factorial of $n$
}}
{{eqn | ll= \leadsto
| l = \dbinom {2 n} n \paren {n!}^2
| r = \paren {2 n}!
| c =
}}
{{end-eqn}}
Let $\mathbb P$ ... | Let $\dbinom {2 n} n$ denote a [[Definition:Binomial Coefficient|binomial coefficient]].
Then for all [[Definition:Prime Number|prime numbers]] $p$ such that $n < p < 2 n$:
:$p \divides \dbinom {2 n} n$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]]. | By definition of [[Definition:Binomial Coefficient|binomial coefficient]]:
{{begin-eqn}}
{{eqn | l = \dbinom {2 n} n
| r = \dfrac {\paren {2 n}!} {\paren {n!}^2}
| c = where $n!$ denotes the [[Definition:Factorial|factorial]] of $n$
}}
{{eqn | ll= \leadsto
| l = \dbinom {2 n} n \paren {n!}^2
| ... | Binomial Coefficient 2 n Choose n is Divisible by All Primes between n and 2 n | https://proofwiki.org/wiki/Binomial_Coefficient_2_n_Choose_n_is_Divisible_by_All_Primes_between_n_and_2_n | https://proofwiki.org/wiki/Binomial_Coefficient_2_n_Choose_n_is_Divisible_by_All_Primes_between_n_and_2_n | [
"Binomial Coefficients"
] | [
"Definition:Binomial Coefficient",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Binomial Coefficient",
"Definition:Factorial",
"Definition:Set",
"Definition:Prime Number",
"Definition:Factorial",
"Prime iff Coprime to all Smaller Positive Integers",
"Definition:Divisor (Algebra)/Integer",
"Euclid's Lemma for Prime Divisors",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-16065 | Path-Connected Set in Subspace | Let $T = \struct {S, \tau}$ be a topological space.
Let $A \subseteq B \subseteq S$.
Let $T_B = \struct {B, \tau_B}$ be the topological space where $\tau_B$ is the subspace topology on $B$.
Then
:$A$ is path-connected in $T_B$ {{iff}} $A$ is path-connected in $T$. | Let $\tau_A$ be the subspace topology on $A$ induced by $\tau$.
Let $\tau'_A$ be the subspace topology on $A$ induced by $\tau_B$.
By the definition of a path-connected set, $A$ is path-connected in $T$ {{iff}} every two points in $A$ are path-connected in $\struct {A, \tau_A}$.
Similarly, $A$ is path-connected in $T_B... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A \subseteq B \subseteq S$.
Let $T_B = \struct {B, \tau_B}$ be the [[Definition:Topological Space|topological space]] where $\tau_B$ is the [[Definition:Subspace Topology|subspace topology]] on $B$.
Then
:$A$ is [[Definition:P... | Let $\tau_A$ be the [[Definition:Subspace Topology|subspace topology]] on $A$ induced by $\tau$.
Let $\tau'_A$ be the [[Definition:Subspace Topology|subspace topology]] on $A$ induced by $\tau_B$.
By the definition of a [[Definition:Path-Connected Set|path-connected set]], $A$ is [[Definition:Path-Connected Set|path-... | Path-Connected Set in Subspace | https://proofwiki.org/wiki/Path-Connected_Set_in_Subspace | https://proofwiki.org/wiki/Path-Connected_Set_in_Subspace | [
"Path-Connected Sets",
"Path-Connected Spaces"
] | [
"Definition:Topological Space",
"Definition:Topological Space",
"Definition:Topological Subspace",
"Definition:Path-Connected/Set",
"Definition:Path-Connected/Set"
] | [
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Definition:Path-Connected/Set",
"Definition:Path-Connected/Set",
"Definition:Path-Connected/Points",
"Definition:Path-Connected/Set",
"Definition:Path-Connected/Points",
"Subspace of Subspace is Subspace",
"Definition:Path-Connec... |
proofwiki-16066 | Precisely One Function in terms of And, Or and Not | Let $\map P {A, B, C}$ denote the precisely one function on the statements $A$, $B$ and $C$.
Then:
:$\map P {A, B, C} \dashv \vdash \paren {A \land \neg B \land \neg C} \lor \paren {\neg A \land B \land \neg C} \lor \paren {\neg A \land \neg B \land C}$
where:
:$\land$ denotes conjunction
:$\lor$ denotes disjunction
:$... | We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.
$\begin{array}{|cccc||ccccc|} \hline
P & (A & B & C) & (((A & \land & \neg & B) & \land & \neg & C) & \lor & ((\neg & A & \land & B) & \land & \neg & C)) & \lor & ((\neg ... | Let $\map P {A, B, C}$ denote the [[Definition:Precisely One Function|precisely one function]] on the [[Definition:Statement|statements]] $A$, $B$ and $C$.
Then:
:$\map P {A, B, C} \dashv \vdash \paren {A \land \neg B \land \neg C} \lor \paren {\neg A \land B \land \neg C} \lor \paren {\neg A \land \neg B \land C}$
wh... | We apply the [[Method of Truth Tables]].
As can be seen by inspection, the [[Definition:Truth Value|truth values]] under the [[Definition:Main Connective (Propositional Logic)|main connectives]] match for all [[Definition:Boolean Interpretation|boolean interpretations]].
$\begin{array}{|cccc||ccccc|} \hline
P & (A &... | Precisely One Function in terms of And, Or and Not | https://proofwiki.org/wiki/Precisely_One_Function_in_terms_of_And,_Or_and_Not | https://proofwiki.org/wiki/Precisely_One_Function_in_terms_of_And,_Or_and_Not | [
"Precisely One Function"
] | [
"Definition:Precisely One Function",
"Definition:Statement",
"Definition:Conjunction",
"Definition:Disjunction",
"Definition:Logical Not"
] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-16067 | Lévy's Continuity Theorem | Let $\sequence {X_n}_{n \mathop \ge 1}$ be a sequence of discrete random variables with characteristic functions $\map {\phi_n} t := E \sqbrk {e^{i t X_n} }$.
Let the sequence $\sequence {\phi_n}$ converge to some real function $\phi$:
:$\forall t \in \R: \map {\phi_n} t \to \map \phi t$.
Then the following statements ... | {{ProofWanted}}
{{Namedfor|Paul Pierre Lévy|cat = Lévy}} | Let $\sequence {X_n}_{n \mathop \ge 1}$ be a [[Definition:Sequence|sequence]] of [[Definition:Discrete Random Variable|discrete random variables]] with [[Definition:Characteristic Function of Random Variable|characteristic functions]] $\map {\phi_n} t := E \sqbrk {e^{i t X_n} }$.
Let the [[Definition:Sequence|sequenc... | {{ProofWanted}}
{{Namedfor|Paul Pierre Lévy|cat = Lévy}} | Lévy's Continuity Theorem | https://proofwiki.org/wiki/Lévy's_Continuity_Theorem | https://proofwiki.org/wiki/Lévy's_Continuity_Theorem | [
"Probability Theory"
] | [
"Definition:Sequence",
"Definition:Random Variable/Discrete",
"Definition:Characteristic Function of Random Variable",
"Definition:Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Real Function",
"Definition:Convergence in Distribution",
"Definition:Characteristic Function of Rand... | [] |
proofwiki-16068 | Composition of Three Mappings which form Identity Mapping | Let $A$, $B$ and $C$ be non-empty sets.
Let $f: A \to B$, $g: B \to C$ and $h: C \to A$ be mappings.
Let the following hold:
{{begin-eqn}}
{{eqn | l = h \circ g \circ f
| r = I_A
}}
{{eqn | l = f \circ h \circ g
| r = I_B
}}
{{eqn | l = g \circ f \circ h
| r = I_C
}}
{{end-eqn}}
where:
:$g \circ f$ (a... | First note that from Composition of Mappings is Associative:
:$\paren {h \circ g} \circ f = h \circ \paren {g \circ f}$
and so on.
However, while there is no need to use parenthesis to establish the order of composition of mappings, in the following the technique will be used in order to clarify what is being done.
We ... | Let $A$, $B$ and $C$ be [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|sets]].
Let $f: A \to B$, $g: B \to C$ and $h: C \to A$ be [[Definition:Mapping|mappings]].
Let the following hold:
{{begin-eqn}}
{{eqn | l = h \circ g \circ f
| r = I_A
}}
{{eqn | l = f \circ h \circ g
| r = I_B
}}
{{eqn | ... | First note that from [[Composition of Mappings is Associative]]:
:$\paren {h \circ g} \circ f = h \circ \paren {g \circ f}$
and so on.
However, while there is no need to use [[Definition:Parenthesis|parenthesis]] to establish the order of [[Definition:Composition of Mappings|composition of mappings]], in the followin... | Composition of Three Mappings which form Identity Mapping | https://proofwiki.org/wiki/Composition_of_Three_Mappings_which_form_Identity_Mapping | https://proofwiki.org/wiki/Composition_of_Three_Mappings_which_form_Identity_Mapping | [
"Bijections",
"Composite Mappings"
] | [
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Mapping",
"Definition:Composition of Mappings",
"Definition:Identity Mapping",
"Definition:Bijection",
"Definition:Inverse Mapping"
] | [
"Composition of Mappings is Associative",
"Definition:Parenthesis",
"Definition:Composition of Mappings",
"Injection iff Left Inverse",
"Definition:Injection",
"Surjection iff Right Inverse",
"Definition:Surjection",
"Definition:Bijection",
"Definition:Bijection",
"Definition:Left Inverse Mapping"... |
proofwiki-16069 | Composition of Product Mappings on Natural Numbers | Let $a \in \N$ be a natural number.
Let $\mu_a: \N \to \N$ be the mapping defined as:
:$\forall x \in \N: \map {\mu_a} x = x a$
Then:
:$\mu_{a b} = \mu_b \circ \mu_a$ | {{begin-eqn}}
{{eqn | l = \mu_{a b}
| r = x \paren {a b}
| c = Definition of $\mu$
}}
{{eqn | r = \paren {x a} b
| c =
}}
{{eqn | r = \paren {\map {\mu_a} x} b
| c = Definition of $\mu$
}}
{{eqn | r = \map {\mu_b} {\map {\mu_a} x}
| c = Definition of $\mu$
}}
{{eqn | r = \map {\paren {\mu... | Let $a \in \N$ be a [[Definition:Natural Number|natural number]].
Let $\mu_a: \N \to \N$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall x \in \N: \map {\mu_a} x = x a$
Then:
:$\mu_{a b} = \mu_b \circ \mu_a$ | {{begin-eqn}}
{{eqn | l = \mu_{a b}
| r = x \paren {a b}
| c = Definition of $\mu$
}}
{{eqn | r = \paren {x a} b
| c =
}}
{{eqn | r = \paren {\map {\mu_a} x} b
| c = Definition of $\mu$
}}
{{eqn | r = \map {\mu_b} {\map {\mu_a} x}
| c = Definition of $\mu$
}}
{{eqn | r = \map {\paren {\mu... | Composition of Product Mappings on Natural Numbers | https://proofwiki.org/wiki/Composition_of_Product_Mappings_on_Natural_Numbers | https://proofwiki.org/wiki/Composition_of_Product_Mappings_on_Natural_Numbers | [
"Composite Mappings",
"Natural Number Multiplication"
] | [
"Definition:Natural Numbers",
"Definition:Mapping"
] | [] |
proofwiki-16070 | Linear Combination of Normal Random Variables | Let $X_1, X_2, X_3, \ldots, X_n$ be independent random variables.
Let $\sequence {\alpha_i}_{1 \mathop \le i \mathop \le n}$ and $\sequence {\mu_i}_{1 \mathop \le i \mathop \le n}$ be sequences of real numbers.
Let $\sequence {\sigma_i}_{1 \mathop \le i \mathop \le n}$ be a sequence of positive real numbers.
Let $X_i ... | Let:
:$\ds Z = \sum_{i \mathop = 1}^n \alpha_i X_i$
Let $M_Z$ be the moment generating function of $Z$.
We aim to show that:
:$\ds Z \sim \Gaussian {\sum_{i \mathop = 1}^n \alpha_i \mu_i} {\sum_{i \mathop = 1}^n \alpha^2_i \sigma^2_i}$
By Moment Generating Function of Normal Distribution and Moment Generating Funct... | Let $X_1, X_2, X_3, \ldots, X_n$ be [[Definition:Independent Random Variables|independent random variables]].
Let $\sequence {\alpha_i}_{1 \mathop \le i \mathop \le n}$ and $\sequence {\mu_i}_{1 \mathop \le i \mathop \le n}$ be [[Definition:Sequence|sequences]] of [[Definition:Real Number|real numbers]].
Let $\seque... | Let:
:$\ds Z = \sum_{i \mathop = 1}^n \alpha_i X_i$
Let $M_Z$ be the [[Definition:Moment Generating Function|moment generating function]] of $Z$.
We aim to show that:
:$\ds Z \sim \Gaussian {\sum_{i \mathop = 1}^n \alpha_i \mu_i} {\sum_{i \mathop = 1}^n \alpha^2_i \sigma^2_i}$
By [[Moment Generating Function o... | Linear Combination of Normal Random Variables | https://proofwiki.org/wiki/Linear_Combination_of_Normal_Random_Variables | https://proofwiki.org/wiki/Linear_Combination_of_Normal_Random_Variables | [
"Normal Distribution"
] | [
"Definition:Independent Random Variables",
"Definition:Sequence",
"Definition:Real Number",
"Definition:Sequence",
"Definition:Positive/Real Number",
"Definition:Normal Distribution"
] | [
"Definition:Moment Generating Function",
"Moment Generating Function of Normal Distribution",
"Moment Generating Function is Unique",
"Moment Generating Function of Normal Distribution",
"Definition:Moment Generating Function",
"Moment Generating Function of Linear Combination of Independent Random Variab... |
proofwiki-16071 | Moment Generating Function of Linear Transformation of Random Variable | Let $X$ be a random variable.
Let $\alpha$ and $\beta$ be real numbers.
Let $Z = \alpha X + \beta$.
Let $M_X$ be the moment generating function of $X$.
Then the moment generating function of $Z$, $M_Z$, is given by:
:$\map {M_Z} t = e^{\beta t} \map {M_X} {\alpha t}$ | {{begin-eqn}}
{{eqn | l = \map {M_Z} t
| r = \expect {\map \exp {t Z} }
| c = {{Defof|Moment Generating Function}}
}}
{{eqn | r = \expect {\map \exp {t \paren {\alpha X + \beta} } }
}}
{{eqn | r = \expect {\map \exp {\paren {\alpha t} X} \map \exp {\beta t} }
| c = Exponential of Sum
}}
{{eqn | r = \m... | Let $X$ be a [[Definition:Random Variable|random variable]].
Let $\alpha$ and $\beta$ be [[Definition:Real Number|real numbers]].
Let $Z = \alpha X + \beta$.
Let $M_X$ be the [[Definition:Moment Generating Function|moment generating function]] of $X$.
Then the [[Definition:Moment Generating Function|moment gener... | {{begin-eqn}}
{{eqn | l = \map {M_Z} t
| r = \expect {\map \exp {t Z} }
| c = {{Defof|Moment Generating Function}}
}}
{{eqn | r = \expect {\map \exp {t \paren {\alpha X + \beta} } }
}}
{{eqn | r = \expect {\map \exp {\paren {\alpha t} X} \map \exp {\beta t} }
| c = [[Exponential of Sum]]
}}
{{eqn | r ... | Moment Generating Function of Linear Transformation of Random Variable | https://proofwiki.org/wiki/Moment_Generating_Function_of_Linear_Transformation_of_Random_Variable | https://proofwiki.org/wiki/Moment_Generating_Function_of_Linear_Transformation_of_Random_Variable | [
"Moment Generating Functions"
] | [
"Definition:Random Variable",
"Definition:Real Number",
"Definition:Moment Generating Function",
"Definition:Moment Generating Function"
] | [
"Exponential of Sum",
"Expectation is Linear",
"Category:Moment Generating Functions"
] |
proofwiki-16072 | Linear Transformation of Normal Random Variable | Let $\mu$, $\alpha$ and $\beta$ be real numbers.
Let $\sigma$ be a positive real number.
Let $X \sim \Gaussian \mu {\sigma^2}$ where $\Gaussian \mu {\sigma^2}$ is the normal distribution with parameters $\mu$ and $\sigma^2$.
Then:
:$\alpha X + \beta \sim \Gaussian {\alpha \mu + \beta} {\alpha^2 \sigma^2}$ | Let $Z = \alpha X + \beta$.
Let $M_Z$ be the moment generating function of $Z$.
We aim to show that:
:$Z \sim \Gaussian {\alpha \mu + \beta} {\alpha^2 \sigma^2}$
By Moment Generating Function of Normal Distribution and Moment Generating Function is Unique, it is sufficient to show that:
:$\map {M_Z} t = \map \exp {\... | Let $\mu$, $\alpha$ and $\beta$ be [[Definition:Real Number|real numbers]].
Let $\sigma$ be a [[Definition:Positive Real Number|positive real number]].
Let $X \sim \Gaussian \mu {\sigma^2}$ where $\Gaussian \mu {\sigma^2}$ is the [[Definition:Normal Distribution|normal distribution]] with parameters $\mu$ and $\sig... | Let $Z = \alpha X + \beta$.
Let $M_Z$ be the [[Definition:Moment Generating Function|moment generating function]] of $Z$.
We aim to show that:
:$Z \sim \Gaussian {\alpha \mu + \beta} {\alpha^2 \sigma^2}$
By [[Moment Generating Function of Normal Distribution]] and [[Moment Generating Function is Unique]], it is ... | Linear Transformation of Normal Random Variable | https://proofwiki.org/wiki/Linear_Transformation_of_Normal_Random_Variable | https://proofwiki.org/wiki/Linear_Transformation_of_Normal_Random_Variable | [
"Normal Distribution"
] | [
"Definition:Real Number",
"Definition:Positive/Real Number",
"Definition:Normal Distribution"
] | [
"Definition:Moment Generating Function",
"Moment Generating Function of Normal Distribution",
"Moment Generating Function is Unique",
"Moment Generating Function of Normal Distribution",
"Definition:Moment Generating Function",
"Moment Generating Function of Linear Combination of Independent Random Variab... |
proofwiki-16073 | Standard Normal Random Variable as Transformation of Normal Random Variable | Let $\mu$ be a real number.
Let $\sigma$ be a positive real number.
Let $X \sim \Gaussian \mu {\sigma^2}$ where $\Gaussian \mu {\sigma^2}$ is the normal distribution with parameters $\mu$ and $\sigma^2$.
Then:
:$\dfrac {X - \mu} \sigma \sim \Gaussian 0 1$
where $\Gaussian 0 1$ is the standard normal distribution. | {{begin-eqn}}
{{eqn | l = \frac {X - \mu} \sigma
| r = \frac 1 \sigma X - \frac \mu \sigma
}}
{{eqn | o = \sim
| r = \Gaussian {\frac \mu \sigma - \frac \mu \sigma} {\paren {\frac 1 \sigma}^2 \sigma^2}
| c = Linear Transformation of Normal Random Variable
}}
{{eqn | r = \Gaussian 0 1
}}
{{end-eqn}}
{{... | Let $\mu$ be a [[Definition:Real Number|real number]].
Let $\sigma$ be a [[Definition:Positive Real Number|positive real number]].
Let $X \sim \Gaussian \mu {\sigma^2}$ where $\Gaussian \mu {\sigma^2}$ is the [[Definition:Normal Distribution|normal distribution]] with parameters $\mu$ and $\sigma^2$.
Then:
:$\... | {{begin-eqn}}
{{eqn | l = \frac {X - \mu} \sigma
| r = \frac 1 \sigma X - \frac \mu \sigma
}}
{{eqn | o = \sim
| r = \Gaussian {\frac \mu \sigma - \frac \mu \sigma} {\paren {\frac 1 \sigma}^2 \sigma^2}
| c = [[Linear Transformation of Normal Random Variable]]
}}
{{eqn | r = \Gaussian 0 1
}}
{{end-eqn}... | Standard Normal Random Variable as Transformation of Normal Random Variable | https://proofwiki.org/wiki/Standard_Normal_Random_Variable_as_Transformation_of_Normal_Random_Variable | https://proofwiki.org/wiki/Standard_Normal_Random_Variable_as_Transformation_of_Normal_Random_Variable | [
"Normal Distribution"
] | [
"Definition:Real Number",
"Definition:Positive/Real Number",
"Definition:Normal Distribution",
"Definition:Standard Normal Distribution"
] | [
"Linear Transformation of Normal Random Variable",
"Category:Normal Distribution"
] |
proofwiki-16074 | Composition of Addition Mappings on Natural Numbers | Let $a \in \N$ be a natural number.
Let $\alpha_a: \N \to \N$ be the mapping defined as:
:$\forall x \in \N: \map {\alpha_a} x = x + a$
Then:
:$\alpha_{a + b} = \alpha_b \circ \alpha_a$ | {{begin-eqn}}
{{eqn | l = \alpha_{a + b}
| r = x + \paren {a + b}
| c = Definition of $\alpha$
}}
{{eqn | r = \paren {x + a} + b
| c =
}}
{{eqn | r = \paren {\map {\alpha_a} x} + b
| c = Definition of $\alpha$
}}
{{eqn | r = \map {\alpha_b} {\map {\alpha_a} x}
| c = Definition of $\alpha$... | Let $a \in \N$ be a [[Definition:Natural Number|natural number]].
Let $\alpha_a: \N \to \N$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall x \in \N: \map {\alpha_a} x = x + a$
Then:
:$\alpha_{a + b} = \alpha_b \circ \alpha_a$ | {{begin-eqn}}
{{eqn | l = \alpha_{a + b}
| r = x + \paren {a + b}
| c = Definition of $\alpha$
}}
{{eqn | r = \paren {x + a} + b
| c =
}}
{{eqn | r = \paren {\map {\alpha_a} x} + b
| c = Definition of $\alpha$
}}
{{eqn | r = \map {\alpha_b} {\map {\alpha_a} x}
| c = Definition of $\alpha$... | Composition of Addition Mappings on Natural Numbers | https://proofwiki.org/wiki/Composition_of_Addition_Mappings_on_Natural_Numbers | https://proofwiki.org/wiki/Composition_of_Addition_Mappings_on_Natural_Numbers | [
"Composite Mappings",
"Natural Number Addition"
] | [
"Definition:Natural Numbers",
"Definition:Mapping"
] | [] |
proofwiki-16075 | Reciprocal of Random Variable with Snedecor's F-Distribution has Snedecor's F-Distribution | Let $n, m$ be strictly positive integers.
Let $X \sim F_{n, m}$ where $F_{n, m}$ is the Snedecor's $F$-distribution with $\tuple {n, m}$ degrees of freedom.
Then:
:$\dfrac 1 X \sim F_{m, n}$ | Let $Y \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.
Let $Z \sim \chi^2_m$ where $\chi^2_m$ is the chi-squared distribution with $m$ degrees of freedom.
Then, by Quotient of Independent Random Variables with Chi-Squared Distribution Divided by Degrees of Freedom has Snede... | Let $n, m$ be [[Definition:Strictly Positive Integer|strictly positive integers]].
Let $X \sim F_{n, m}$ where $F_{n, m}$ is the [[Definition:Snedecor's F-Distribution|Snedecor's $F$-distribution]] with $\tuple {n, m}$ degrees of freedom.
Then:
:$\dfrac 1 X \sim F_{m, n}$ | Let $Y \sim \chi^2_n$ where $\chi^2_n$ is the [[Definition:Chi-Squared Distribution|chi-squared distribution]] with $n$ degrees of freedom.
Let $Z \sim \chi^2_m$ where $\chi^2_m$ is the [[Definition:Chi-Squared Distribution|chi-squared distribution]] with $m$ degrees of freedom.
Then, by [[Quotient of Independent Ra... | Reciprocal of Random Variable with Snedecor's F-Distribution has Snedecor's F-Distribution | https://proofwiki.org/wiki/Reciprocal_of_Random_Variable_with_Snedecor's_F-Distribution_has_Snedecor's_F-Distribution | https://proofwiki.org/wiki/Reciprocal_of_Random_Variable_with_Snedecor's_F-Distribution_has_Snedecor's_F-Distribution | [
"Snedecor's F-Distribution"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Snedecor's F-Distribution"
] | [
"Definition:Chi-Squared Distribution",
"Definition:Chi-Squared Distribution",
"Quotient of Independent Random Variables with Chi-Squared Distribution Divided by Degrees of Freedom has Snedecor's F-Distribution",
"Quotient of Independent Random Variables with Chi-Squared Distribution Divided by Degrees of Free... |
proofwiki-16076 | Quotient of Independent Random Variables with Chi-Squared Distribution Divided by Degrees of Freedom has Snedecor's F-Distribution | Let $n$ and $m$ be strictly positive integers.
Let $X$ and $Y$ be independent random variables.
Let $X \sim \chi_n^2$ where $\chi_n^2$ is the chi-squared distribution with $n$ degrees of freedom.
Let $Y \sim \chi_m^2$ where $\chi_m^2$ is the chi-squared distribution with $m$ degrees of freedom.
Then:
:$\dfrac {X / n... | {{ProofWanted}}
Category:Chi-Squared Distribution
Category:Snedecor's F-Distribution
8gywnfks3vlnwzzbkg6kpmj4thftam6 | Let $n$ and $m$ be [[Definition:Strictly Positive Integer|strictly positive integers]].
Let $X$ and $Y$ be [[Definition:Independent Random Variables|independent random variables]].
Let $X \sim \chi_n^2$ where $\chi_n^2$ is the [[Definition:Chi-Squared Distribution|chi-squared distribution]] with $n$ degrees of freed... | {{ProofWanted}}
[[Category:Chi-Squared Distribution]]
[[Category:Snedecor's F-Distribution]]
8gywnfks3vlnwzzbkg6kpmj4thftam6 | Quotient of Independent Random Variables with Chi-Squared Distribution Divided by Degrees of Freedom has Snedecor's F-Distribution | https://proofwiki.org/wiki/Quotient_of_Independent_Random_Variables_with_Chi-Squared_Distribution_Divided_by_Degrees_of_Freedom_has_Snedecor's_F-Distribution | https://proofwiki.org/wiki/Quotient_of_Independent_Random_Variables_with_Chi-Squared_Distribution_Divided_by_Degrees_of_Freedom_has_Snedecor's_F-Distribution | [
"Chi-Squared Distribution",
"Snedecor's F-Distribution"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Independent Random Variables",
"Definition:Chi-Squared Distribution",
"Definition:Chi-Squared Distribution",
"Definition:Snedecor's F-Distribution"
] | [
"Category:Chi-Squared Distribution",
"Category:Snedecor's F-Distribution"
] |
proofwiki-16077 | Square of Random Variable with t-Distribution has Snedecor's F-Distribution | Let $k$ be a strictly positive integer.
Let $X \sim t_k$ where $t_k$ is the $t$-distribution with $k$ degrees of freedom.
Then:
:$X^2 \sim F_{1, k}$
where $F_{1, k}$ is Snedecor's $F$-distribution with $\tuple {1, k}$ degrees of freedom. | Let $Y \sim F_{1, k}$.
We aim to show that:
:$\map \Pr {Y < x^2} = \map \Pr {\size X < x}$
for all $x \ge 0$.
That is:
:$\map \Pr {Y < x^2} = \map \Pr {-x < X < x}$
for all $x \ge 0$.
We have:
{{begin-eqn}}
{{eqn | l = \map \Pr {Y < x^2}
| r = \int_0^{x^2} \frac {k^{k/2} 1^{1/2} u^{\paren {1/2} - 1} } {\par... | Let $k$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $X \sim t_k$ where $t_k$ is the [[Definition:Student's t-Distribution|$t$-distribution]] with $k$ [[Definition:Degrees of Freedom of Student's t-Distribution|degrees of freedom]].
Then:
:$X^2 \sim F_{1, k}$
where $F_{1, k}$ is [... | Let $Y \sim F_{1, k}$.
We aim to show that:
:$\map \Pr {Y < x^2} = \map \Pr {\size X < x}$
for all $x \ge 0$.
That is:
:$\map \Pr {Y < x^2} = \map \Pr {-x < X < x}$
for all $x \ge 0$.
We have:
{{begin-eqn}}
{{eqn | l = \map \Pr {Y < x^2}
| r = \int_0^{x^2} \frac {k^{k/2} 1^{1/2} u^{\paren {1/2} - 1}... | Square of Random Variable with t-Distribution has Snedecor's F-Distribution | https://proofwiki.org/wiki/Square_of_Random_Variable_with_t-Distribution_has_Snedecor's_F-Distribution | https://proofwiki.org/wiki/Square_of_Random_Variable_with_t-Distribution_has_Snedecor's_F-Distribution | [
"Student's t-Distribution",
"Snedecor's F-Distribution"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Student's t-Distribution",
"Definition:Student's t-Distribution/Degrees of Freedom",
"Definition:Snedecor's F-Distribution",
"Definition:Degrees of Freedom"
] | [
"Gamma Function of One Half",
"Integration by Substitution",
"Definite Integral of Even Function"
] |
proofwiki-16078 | Union of Connected Sets with Common Point is Connected | Let $T = \struct {S, \tau}$ be a topological space.
Let $\family {B_\alpha}_{\alpha \mathop \in A}$ be a family of connected sets of $T$.
Let $\exists x \in \ds \bigcap \family {B_\alpha}_{\alpha \mathop \in A}$.
Then
:$\ds \bigcup \family {B_\alpha}_{\alpha \mathop \in A}$ is a connected set of $T$. | Let $B = \ds \bigcup_{\alpha \mathop \in A} B_\alpha$.
{{Recall|Connected Set (Topology)|connected set|index = 3}}
{{:Definition:Connected Set (Topology)/Definition 3}}
Thus $B$ is connected in $T$ {{iff}} the subspace $\struct {B, \tau_B}$ is a connected space.
{{Recall|Connected Topological Space|connected space|inde... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\family {B_\alpha}_{\alpha \mathop \in A}$ be a [[Definition:Indexed Family|family]] of [[Definition:Connected Set (Topology)|connected sets]] of $T$.
Let $\exists x \in \ds \bigcap \family {B_\alpha}_{\alpha \mathop \in A}$.
... | Let $B = \ds \bigcup_{\alpha \mathop \in A} B_\alpha$.
{{Recall|Connected Set (Topology)|connected set|index = 3}}
{{:Definition:Connected Set (Topology)/Definition 3}}
Thus $B$ is [[Definition:Connected Set (Topology)|connected]] in $T$ {{iff}} the [[Definition:Topological Subspace|subspace]] $\struct {B, \tau_B}$ ... | Union of Connected Sets with Common Point is Connected/Proof 1 | https://proofwiki.org/wiki/Union_of_Connected_Sets_with_Common_Point_is_Connected | https://proofwiki.org/wiki/Union_of_Connected_Sets_with_Common_Point_is_Connected/Proof_1 | [
"Union of Connected Sets with Common Point is Connected",
"Connected Sets (Topology)"
] | [
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Connected Set (Topology)",
"Definition:Connected Set (Topology)"
] | [
"Definition:Connected Set (Topology)",
"Definition:Topological Subspace",
"Definition:Connected Topological Space",
"Definition:Connected Topological Space",
"Definition:Clopen Set",
"Definition:Clopen Set",
"Definition:Topological Subspace",
"Complement of Clopen Set is Clopen",
"Definition:Clopen ... |
proofwiki-16079 | Union of Connected Sets with Common Point is Connected | Let $T = \struct {S, \tau}$ be a topological space.
Let $\family {B_\alpha}_{\alpha \mathop \in A}$ be a family of connected sets of $T$.
Let $\exists x \in \ds \bigcap \family {B_\alpha}_{\alpha \mathop \in A}$.
Then
:$\ds \bigcup \family {B_\alpha}_{\alpha \mathop \in A}$ is a connected set of $T$. | Follows immediately from Union of Connected Sets with Non-Empty Intersections is Connected.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\family {B_\alpha}_{\alpha \mathop \in A}$ be a [[Definition:Indexed Family|family]] of [[Definition:Connected Set (Topology)|connected sets]] of $T$.
Let $\exists x \in \ds \bigcap \family {B_\alpha}_{\alpha \mathop \in A}$.
... | Follows immediately from [[Union of Connected Sets with Non-Empty Intersections is Connected]].
{{qed}} | Union of Connected Sets with Common Point is Connected/Proof 2 | https://proofwiki.org/wiki/Union_of_Connected_Sets_with_Common_Point_is_Connected | https://proofwiki.org/wiki/Union_of_Connected_Sets_with_Common_Point_is_Connected/Proof_2 | [
"Union of Connected Sets with Common Point is Connected",
"Connected Sets (Topology)"
] | [
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Connected Set (Topology)",
"Definition:Connected Set (Topology)"
] | [
"Union of Connected Sets with Non-Empty Intersections is Connected"
] |
proofwiki-16080 | Connected Subset of Union of Disjoint Open Sets | Let $T = \struct{S, \tau}$ be a topological space.
Let $A$ be a connected set of $T$.
Let $U, V$ be disjoint open sets.
Let $A \subseteq U \cup V$.
Then
:either $A \subseteq U$ or $A \subseteq V$. | Let $U' = A \cap U$ and $V' = A \cap V$.
By definition $U'$ and $V'$ are open sets in the subspace $\struct{A, \tau_A}$.
From Intersection is Empty Implies Intersection of Subsets is Empty $U'$ and $V'$ are disjoint.
Hence $U'$ and $V'$ are separated sets by definition.
Now
{{begin-eqn}}
{{eqn | l = A
| r = A \c... | Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A$ be a [[Definition:Connected Set (Topology)|connected set]] of $T$.
Let $U, V$ be [[Definition:Disjoint Sets|disjoint]] [[Definition:Open Set|open sets]].
Let $A \subseteq U \cup V$.
Then
:either $A \subseteq U$ or $A \subse... | Let $U' = A \cap U$ and $V' = A \cap V$.
By definition $U'$ and $V'$ are [[Definition:Open Set|open sets]] in the [[Definition:Topological Subspace|subspace]] $\struct{A, \tau_A}$.
From [[Intersection is Empty Implies Intersection of Subsets is Empty]] $U'$ and $V'$ are [[Definition:Disjoint Sets|disjoint]].
Hence ... | Connected Subset of Union of Disjoint Open Sets | https://proofwiki.org/wiki/Connected_Subset_of_Union_of_Disjoint_Open_Sets | https://proofwiki.org/wiki/Connected_Subset_of_Union_of_Disjoint_Open_Sets | [
"Connected Topological Spaces"
] | [
"Definition:Topological Space",
"Definition:Connected Set (Topology)",
"Definition:Disjoint Sets",
"Definition:Open Set"
] | [
"Definition:Open Set",
"Definition:Topological Subspace",
"Set Intersection Preserves Subsets/Families of Sets/Intersection is Empty Implies Intersection of Subsets is Empty",
"Definition:Disjoint Sets",
"Definition:Separated Sets",
"Intersection with Subset is Subset",
"Intersection Distributes over Un... |
proofwiki-16081 | Axiom of Archimedes/Variant | Let $x$ and $y$ be a natural numbers.
Then there exists a natural number $n$ such that:
:$n x \ge y$ | {{AimForCont}} there exists $x, y \in \N$ such that $n x < y$ for every natural number $n$.
Consider the set $S$, defined as:
:$S := \set {y - n x: n \in \N}$
We have {{hypothesis}} that $S$ contains only natural numbers.
By the Well-Ordering Principle, $S$ contains a smallest element, $y - m x$ for example.
But $y - \... | Let $x$ and $y$ be a [[Definition:Natural Numbers|natural numbers]].
Then there exists a [[Definition:Natural Numbers|natural number]] $n$ such that:
:$n x \ge y$ | {{AimForCont}} there exists $x, y \in \N$ such that $n x < y$ for every [[Definition:Natural Numbers|natural number]] $n$.
Consider the [[Definition:Set|set]] $S$, defined as:
:$S := \set {y - n x: n \in \N}$
We have {{hypothesis}} that $S$ contains only [[Definition:Natural Numbers|natural numbers]].
By the [[Well... | Axiom of Archimedes/Variant | https://proofwiki.org/wiki/Axiom_of_Archimedes/Variant | https://proofwiki.org/wiki/Axiom_of_Archimedes/Variant | [
"Axiom of Archimedes"
] | [
"Definition:Natural Numbers",
"Definition:Natural Numbers"
] | [
"Definition:Natural Numbers",
"Definition:Set",
"Definition:Natural Numbers",
"Well-Ordering Principle",
"Definition:Smallest Element",
"Definition:Natural Numbers",
"Definition:Contradiction",
"Definition:Smallest Element",
"Definition:Natural Numbers",
"Proof by Contradiction"
] |
proofwiki-16082 | Set Intersection Preserves Subsets/Families of Sets/Intersection is Empty Implies Intersection of Subsets is Empty | Let $I$ be an indexing set.
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\alpha}_{\alpha \mathop \in I}$ be indexed families of subsets of a set $S$.
Let:
:$\forall \beta \in I: A_\beta \subseteq B_\beta$
Then:
:$\ds \bigcap_{\alpha \mathop \in I} B_\alpha = \O \implies \bigcap_{\alpha \mathop \in I}... | Let $\ds \bigcap_{\alpha \mathop \in I} B_\alpha = \O$.
From Set Intersection Preserves Subsets/Families of Sets:
:$\ds \bigcap_{\alpha \mathop \in I} A_\alpha \subseteq \bigcap_{\alpha \mathop \in I} B_\alpha = \O$
From Subset of Empty Set:
:$\ds \bigcap_{\alpha \mathop \in I} A_\alpha = \O$
{{qed}}
Category:Set Inter... | Let $I$ be an [[Definition:Indexing Set|indexing set]].
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\alpha}_{\alpha \mathop \in I}$ be [[Definition:Indexed Family of Subsets|indexed families of subsets]] of a [[Definition:Set|set]] $S$.
Let:
:$\forall \beta \in I: A_\beta \subseteq B_\beta$
Then... | Let $\ds \bigcap_{\alpha \mathop \in I} B_\alpha = \O$.
From [[Set Intersection Preserves Subsets/Families of Sets]]:
:$\ds \bigcap_{\alpha \mathop \in I} A_\alpha \subseteq \bigcap_{\alpha \mathop \in I} B_\alpha = \O$
From [[Subset of Empty Set]]:
:$\ds \bigcap_{\alpha \mathop \in I} A_\alpha = \O$
{{qed}}
[[Categ... | Set Intersection Preserves Subsets/Families of Sets/Intersection is Empty Implies Intersection of Subsets is Empty | https://proofwiki.org/wiki/Set_Intersection_Preserves_Subsets/Families_of_Sets/Intersection_is_Empty_Implies_Intersection_of_Subsets_is_Empty | https://proofwiki.org/wiki/Set_Intersection_Preserves_Subsets/Families_of_Sets/Intersection_is_Empty_Implies_Intersection_of_Subsets_is_Empty | [
"Set Intersection Preserves Subsets",
"Indexed Families"
] | [
"Definition:Indexing Set",
"Definition:Indexing Set/Family of Subsets",
"Definition:Set"
] | [
"Set Intersection Preserves Subsets/Families of Sets",
"Subset of Empty Set",
"Category:Set Intersection Preserves Subsets",
"Category:Indexed Families"
] |
proofwiki-16083 | Union of Path-Connected Sets with Common Point is Path-Connected | Let $T = \struct {S, \tau}$ be a topological space.
Let $\family {B_\alpha}_{\alpha \mathop \in A}$ be a family of path-connected sets of $T$.
Let $\exists x \in \ds \bigcap \family {B_\alpha}_{\alpha \mathop \in A}$.
Then
:$\ds \bigcup \family {B_\alpha}_{\alpha \mathop \in A}$ is a path-connected set of $T$. | Let $B = \ds \bigcup \family {B_\alpha}_{\alpha \mathop \in A}$.
Let $a, b \in B$.
Then
:$\exists \alpha, \beta \in A: a \in B_\alpha \land b \in B_\beta$.
As $B_\alpha$ is a path-connected set in $T$ then $a$ and $x$ are path-connected points.
Similarly, $x$ and $b$ are path-connected points.
From Joining Paths makes... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\family {B_\alpha}_{\alpha \mathop \in A}$ be a [[Definition:Indexed Family|family]] of [[Definition:Path-Connected Set|path-connected sets]] of $T$.
Let $\exists x \in \ds \bigcap \family {B_\alpha}_{\alpha \mathop \in A}$.
T... | Let $B = \ds \bigcup \family {B_\alpha}_{\alpha \mathop \in A}$.
Let $a, b \in B$.
Then
:$\exists \alpha, \beta \in A: a \in B_\alpha \land b \in B_\beta$.
As $B_\alpha$ is a [[Definition:Path-Connected Set|path-connected set]] in $T$ then $a$ and $x$ are [[Definition:Path-Connected Points|path-connected points]].
... | Union of Path-Connected Sets with Common Point is Path-Connected | https://proofwiki.org/wiki/Union_of_Path-Connected_Sets_with_Common_Point_is_Path-Connected | https://proofwiki.org/wiki/Union_of_Path-Connected_Sets_with_Common_Point_is_Path-Connected | [
"Path-Connected Sets"
] | [
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Path-Connected/Set",
"Definition:Path-Connected/Set"
] | [
"Definition:Path-Connected/Set",
"Definition:Path-Connected/Points",
"Definition:Path-Connected/Points",
"Joining Paths makes Another Path",
"Definition:Path-Connected/Points",
"Definition:Path-Connected/Set"
] |
proofwiki-16084 | Sum of Powers of 2 | Let $n \in \N_{>0}$ be a (strictly positive) natural number.
Then:
{{begin-eqn}}
{{eqn | n = 1
| l = 2^n - 1
| r = \sum_{j \mathop = 0}^{n - 1} 2^j
| c =
}}
{{eqn | r = 1 + 2 + 2^2 + 2^3 + \dotsb + 2^{n - 1}
| c =
}}
{{end-eqn}} | From Sum of Geometric Sequence:
:$\ds \sum_{j \mathop = 0}^{n - 1} x^j = \frac {x^n - 1} {x - 1}$
The result follows by setting $x = 2$.
{{Qed}} | Let $n \in \N_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly positive)]] [[Definition:Natural Number|natural number]].
Then:
{{begin-eqn}}
{{eqn | n = 1
| l = 2^n - 1
| r = \sum_{j \mathop = 0}^{n - 1} 2^j
| c =
}}
{{eqn | r = 1 + 2 + 2^2 + 2^3 + \dotsb + 2^{n - 1}
| c =
}}
{{end... | From [[Sum of Geometric Sequence]]:
:$\ds \sum_{j \mathop = 0}^{n - 1} x^j = \frac {x^n - 1} {x - 1}$
The result follows by setting $x = 2$.
{{Qed}} | Sum of Powers of 2/Proof 1 | https://proofwiki.org/wiki/Sum_of_Powers_of_2 | https://proofwiki.org/wiki/Sum_of_Powers_of_2/Proof_1 | [
"Sum of Powers of 2",
"Geometric Sequences"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Natural Numbers"
] | [
"Sum of Geometric Sequence"
] |
proofwiki-16085 | Sum of Powers of 2 | Let $n \in \N_{>0}$ be a (strictly positive) natural number.
Then:
{{begin-eqn}}
{{eqn | n = 1
| l = 2^n - 1
| r = \sum_{j \mathop = 0}^{n - 1} 2^j
| c =
}}
{{eqn | r = 1 + 2 + 2^2 + 2^3 + \dotsb + 2^{n - 1}
| c =
}}
{{end-eqn}} | Let $S \subseteq \N_{>0}$ denote the set of (strictly positive) natural numbers for which $(1)$ holds.
=== Basis for the Induction ===
We have:
{{begin-eqn}}
{{eqn | l = 2^1 - 1
| r = 2 - 1
| c =
}}
{{eqn | r = 1
| c =
}}
{{eqn | r = 2^0
| c =
}}
{{eqn | r = \sum_{j \mathop = 0}^{1 - 1} 2^j
... | Let $n \in \N_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly positive)]] [[Definition:Natural Number|natural number]].
Then:
{{begin-eqn}}
{{eqn | n = 1
| l = 2^n - 1
| r = \sum_{j \mathop = 0}^{n - 1} 2^j
| c =
}}
{{eqn | r = 1 + 2 + 2^2 + 2^3 + \dotsb + 2^{n - 1}
| c =
}}
{{end... | Let $S \subseteq \N_{>0}$ denote the [[Definition:set|set]] of [[Definition:Strictly Positive Integer|(strictly positive)]] [[Definition:Natural Number|natural numbers]] for which $(1)$ holds.
=== Basis for the Induction ===
We have:
{{begin-eqn}}
{{eqn | l = 2^1 - 1
| r = 2 - 1
| c =
}}
{{eqn | r = 1
... | Sum of Powers of 2/Proof 2 | https://proofwiki.org/wiki/Sum_of_Powers_of_2 | https://proofwiki.org/wiki/Sum_of_Powers_of_2/Proof_2 | [
"Sum of Powers of 2",
"Geometric Sequences"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Natural Numbers"
] | [
"Definition:set",
"Definition:Strictly Positive/Integer",
"Definition:Natural Numbers",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Sum of Powers of 2/Proof 2",
"Principle of Finite Induction"
] |
proofwiki-16086 | Upper Bound for Lucas Number | Let $L_n$ denote the $n$th Lucas number.
Then:
:$L_n < \paren {\dfrac 7 4}^n$ | The proof proceeds by complete induction.
For all $n \in \Z_{\ge 1}$, let $\map P n$ be the proposition:
:$L_n < \paren {\dfrac 7 4}^n$
$\map P 1$ is the case:
{{begin-eqn}}
{{eqn | l = L_1
| r = 1
| c =
}}
{{eqn | o = <
| r = \dfrac 7 4
| c =
}}
{{end-eqn}}
Thus $\map P 1$ is seen to hold. | Let $L_n$ denote the $n$th [[Definition:Lucas Number|Lucas number]].
Then:
:$L_n < \paren {\dfrac 7 4}^n$ | The proof proceeds by [[Second Principle of Mathematical Induction|complete induction]].
For all $n \in \Z_{\ge 1}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$L_n < \paren {\dfrac 7 4}^n$
$\map P 1$ is the case:
{{begin-eqn}}
{{eqn | l = L_1
| r = 1
| c =
}}
{{eqn | o = <
| ... | Upper Bound for Lucas Number | https://proofwiki.org/wiki/Upper_Bound_for_Lucas_Number | https://proofwiki.org/wiki/Upper_Bound_for_Lucas_Number | [
"Lucas Numbers",
"Proofs by Induction"
] | [
"Definition:Lucas Number"
] | [
"Second Principle of Mathematical Induction",
"Definition:Proposition",
"Second Principle of Mathematical Induction"
] |
proofwiki-16087 | Factorial Greater than Square for n Greater than 3 | Let $n \in \Z$ be an integer such that $n > 3$.
Then $n! > n^2$. | We note that:
{{begin-eqn}}
{{eqn | l = 1!
| r = 1
| c =
}}
{{eqn | r = 1^2
| c =
}}
{{eqn | l = 2!
| r = 2
| c =
}}
{{eqn | o = <
| r = 4
| c =
}}
{{eqn | r = 2^2
| c =
}}
{{eqn | l = 3!
| r = 6
| c =
}}
{{eqn | o = <
| r = 9
| c =
}}
{{eqn... | Let $n \in \Z$ be an [[Definition:Integer|integer]] such that $n > 3$.
Then $n! > n^2$. | We note that:
{{begin-eqn}}
{{eqn | l = 1!
| r = 1
| c =
}}
{{eqn | r = 1^2
| c =
}}
{{eqn | l = 2!
| r = 2
| c =
}}
{{eqn | o = <
| r = 4
| c =
}}
{{eqn | r = 2^2
| c =
}}
{{eqn | l = 3!
| r = 6
| c =
}}
{{eqn | o = <
| r = 9
| c =
}}
{{eq... | Factorial Greater than Square for n Greater than 3 | https://proofwiki.org/wiki/Factorial_Greater_than_Square_for_n_Greater_than_3 | https://proofwiki.org/wiki/Factorial_Greater_than_Square_for_n_Greater_than_3 | [
"Factorials",
"Square Numbers"
] | [
"Definition:Integer"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-16088 | Sum of Sequence of k x k! | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^n j \times j!
| r = 1 \times 1! + 2 \times 2! + 3 \times 3! + \dotsb + n \times n!
| c =
}}
{{eqn | r = \paren {n + 1}! - 1
| c =
}}
{{end-eqn}} | The proof proceeds by induction.
For all $n \in \Z_{\ge 1}$, let $\map P n$ be the proposition:
:$\ds \sum_{j \mathop = 1}^n j \times j! = \paren {n + 1}! - 1$
=== Basis for the Induction ===
$\map P 1$ is the case:
{{begin-eqn}}
{{eqn | l = 1 \times 1!
| r = 1
| c =
}}
{{eqn | r = \paren {1 + 1}! - 1
... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^n j \times j!
| r = 1 \times 1! + 2 \times 2! + 3 \times 3! + \dotsb + n \times n!
| c =
}}
{{eqn | r = \paren {n + 1}! - 1
| c =
}}
{{end-eqn}} | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 1}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \sum_{j \mathop = 1}^n j \times j! = \paren {n + 1}! - 1$
=== Basis for the Induction ===
$\map P 1$ is the case:
{{begin-eqn}}
{{eqn | l = 1 \ti... | Sum of Sequence of k x k!/Proof 1 | https://proofwiki.org/wiki/Sum_of_Sequence_of_k_x_k! | https://proofwiki.org/wiki/Sum_of_Sequence_of_k_x_k!/Proof_1 | [
"Factorials",
"Sum of Sequence of k x k!"
] | [
"Definition:Strictly Positive/Integer"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Sum of Sequence of k x k!",
"Principle of Mathematical Induction"
] |
proofwiki-16089 | Sum of Sequence of k x k! | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^n j \times j!
| r = 1 \times 1! + 2 \times 2! + 3 \times 3! + \dotsb + n \times n!
| c =
}}
{{eqn | r = \paren {n + 1}! - 1
| c =
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^n j \times j!
| r = \sum_{j \mathop = 1}^n \paren {j + 1 - 1} \times j!
}}
{{eqn | r = \sum_{j \mathop = 1}^n \paren {\paren {j + 1} j! - j!}
}}
{{eqn | r = \sum_{j \mathop = 1}^n \paren {\paren {j + 1}! - j!}
}}
{{eqn | r = \paren {n + 1}! - 1!
| c = Telescoping Series:... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^n j \times j!
| r = 1 \times 1! + 2 \times 2! + 3 \times 3! + \dotsb + n \times n!
| c =
}}
{{eqn | r = \paren {n + 1}! - 1
| c =
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^n j \times j!
| r = \sum_{j \mathop = 1}^n \paren {j + 1 - 1} \times j!
}}
{{eqn | r = \sum_{j \mathop = 1}^n \paren {\paren {j + 1} j! - j!}
}}
{{eqn | r = \sum_{j \mathop = 1}^n \paren {\paren {j + 1}! - j!}
}}
{{eqn | r = \paren {n + 1}! - 1!
| c = [[Telescoping Serie... | Sum of Sequence of k x k!/Proof 2 | https://proofwiki.org/wiki/Sum_of_Sequence_of_k_x_k! | https://proofwiki.org/wiki/Sum_of_Sequence_of_k_x_k!/Proof_2 | [
"Factorials",
"Sum of Sequence of k x k!"
] | [
"Definition:Strictly Positive/Integer"
] | [
"Telescoping Series/Example 1"
] |
proofwiki-16090 | Condition for Increasing Binomial Coefficients | Let $n \in \Z_{> 0}$ be a (strictly) positive integer.
Let $\dbinom n k$ denote a binomial coefficient for $k \in \N$.
Then:
:$\dbinom n k < \dbinom n {k + 1} \iff 0 \le k < \dfrac {n - 1} 2$ | {{begin-eqn}}
{{eqn | l = \dbinom n k
| o = <
| r = \dbinom n {k + 1}
}}
{{eqn | ll= \leadstoandfrom
| l = \frac {n!} {\paren {n - k}! k!}
| o = <
| r = \frac {n!} {\paren {n - k - 1}! \paren {k + 1}!}
| c = {{Defof|Binomial Coefficient}}
}}
{{eqn | ll= \leadstoandfrom
| l = k ... | Let $n \in \Z_{> 0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $\dbinom n k$ denote a [[Definition:Binomial Coefficient|binomial coefficient]] for $k \in \N$.
Then:
:$\dbinom n k < \dbinom n {k + 1} \iff 0 \le k < \dfrac {n - 1} 2$ | {{begin-eqn}}
{{eqn | l = \dbinom n k
| o = <
| r = \dbinom n {k + 1}
}}
{{eqn | ll= \leadstoandfrom
| l = \frac {n!} {\paren {n - k}! k!}
| o = <
| r = \frac {n!} {\paren {n - k - 1}! \paren {k + 1}!}
| c = {{Defof|Binomial Coefficient}}
}}
{{eqn | ll= \leadstoandfrom
| l = k ... | Condition for Increasing Binomial Coefficients/Proof 1 | https://proofwiki.org/wiki/Condition_for_Increasing_Binomial_Coefficients | https://proofwiki.org/wiki/Condition_for_Increasing_Binomial_Coefficients/Proof_1 | [
"Binomial Coefficients",
"Condition for Increasing Binomial Coefficients"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Binomial Coefficient"
] | [] |
proofwiki-16091 | Condition for Increasing Binomial Coefficients | Let $n \in \Z_{> 0}$ be a (strictly) positive integer.
Let $\dbinom n k$ denote a binomial coefficient for $k \in \N$.
Then:
:$\dbinom n k < \dbinom n {k + 1} \iff 0 \le k < \dfrac {n - 1} 2$ | The proof proceeds by induction on $n$.
For all $n \in \Z_{> 0}$, let $\map P n$ be the proposition:
:$\dbinom n k < \dbinom n {k + 1} \iff 0 \le k < \dfrac {n - 1} 2$
First we investigate the edge case.
Let $n = 1$.
Then we have:
{{begin-eqn}}
{{eqn | l = \dbinom 1 0
| r = 1
| c = Binomial Coefficient with... | Let $n \in \Z_{> 0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $\dbinom n k$ denote a [[Definition:Binomial Coefficient|binomial coefficient]] for $k \in \N$.
Then:
:$\dbinom n k < \dbinom n {k + 1} \iff 0 \le k < \dfrac {n - 1} 2$ | The proof proceeds by [[Principle of Mathematical Induction|induction]] on $n$.
For all $n \in \Z_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\dbinom n k < \dbinom n {k + 1} \iff 0 \le k < \dfrac {n - 1} 2$
First we investigate the edge case.
Let $n = 1$.
Then we have:
{{begin-eqn}}
{{... | Condition for Increasing Binomial Coefficients/Proof 2 | https://proofwiki.org/wiki/Condition_for_Increasing_Binomial_Coefficients | https://proofwiki.org/wiki/Condition_for_Increasing_Binomial_Coefficients/Proof_2 | [
"Binomial Coefficients",
"Condition for Increasing Binomial Coefficients"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Binomial Coefficient"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Binomial Coefficient with Zero",
"Binomial Coefficient with Self",
"Binomial Coefficient with Zero",
"Binomial Coefficient with One",
"Binomial Coefficient with Self",
"Definition:Basis for the Induction",
"Definition:Induction Hypoth... |
proofwiki-16092 | Condition for Equality of Adjacent Binomial Coefficients | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $\dbinom n k$ denote a binomial coefficient for $k \in \Z$.
Then:
:$\dbinom n k = \dbinom n {k + 1}$
{{iff}}:
:$n$ is an odd integer
:$k = \dfrac {n - 1} 2$ | === Sufficient Condition ===
Let $n$ be odd and $k = \dfrac {n - 1} 2$.
Let $n = 2 m + 1$ for some $m \in \Z_{\ge 0}$.
We have:
{{begin-eqn}}
{{eqn | l = k
| r = \dfrac {n - 1} 2
| c =
}}
{{eqn | r = \dfrac {\paren {2 m + 1} - 1} 2
| c =
}}
{{eqn | r = \dfrac {2 m} 2
| c =
}}
{{eqn | r = m
... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $\dbinom n k$ denote a [[Definition:Binomial Coefficient|binomial coefficient]] for $k \in \Z$.
Then:
:$\dbinom n k = \dbinom n {k + 1}$
{{iff}}:
:$n$ is an [[Definition:Odd Integer|odd integer]]
:$k = \dfrac {n - 1} 2... | === Sufficient Condition ===
Let $n$ be [[Definition:Odd Integer|odd]] and $k = \dfrac {n - 1} 2$.
Let $n = 2 m + 1$ for some $m \in \Z_{\ge 0}$.
We have:
{{begin-eqn}}
{{eqn | l = k
| r = \dfrac {n - 1} 2
| c =
}}
{{eqn | r = \dfrac {\paren {2 m + 1} - 1} 2
| c =
}}
{{eqn | r = \dfrac {2 m} 2
... | Condition for Equality of Adjacent Binomial Coefficients | https://proofwiki.org/wiki/Condition_for_Equality_of_Adjacent_Binomial_Coefficients | https://proofwiki.org/wiki/Condition_for_Equality_of_Adjacent_Binomial_Coefficients | [
"Binomial Coefficients"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Binomial Coefficient",
"Definition:Odd Integer"
] | [
"Definition:Odd Integer",
"Symmetry Rule for Binomial Coefficients",
"Symmetry Rule for Binomial Coefficients",
"Definition:Odd Integer"
] |
proofwiki-16093 | Equivalence of Definitions of Component/Equivalence Class equals Union of Connected Sets | Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in T$.
Let $\CC_x = \set {A \subseteq S: x \in A \land A \text{ is connected in } T}$
Let $C = \bigcup \CC_x$
Let $\sim$ be the equivalence relation defined by:
:$y \sim z$ {{iff}} $y$ and $z$ are connected in $T$.
Let $C'$ be the equivalence class of $\sim$ c... | {{begin-eqn}}
{{eqn | l = y \in C'
| o = \leadstoandfrom
| r = \exists B \text{ a connected set of } T, x \in B, y \in B
| c = Definition of $\sim$
}}
{{eqn | o = \leadstoandfrom
| r = \exists B \in \CC_x : y \in B
| c = equivalent definition
}}
{{eqn | o = \leadstoandfrom
| r = y \i... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in T$.
Let $\CC_x = \set {A \subseteq S: x \in A \land A \text{ is connected in } T}$
Let $C = \bigcup \CC_x$
Let $\sim$ be the [[Definition:Equivalence Relation|equivalence relation]] defined by:
:$y \sim z$ {{iff}} $y$ ... | {{begin-eqn}}
{{eqn | l = y \in C'
| o = \leadstoandfrom
| r = \exists B \text{ a connected set of } T, x \in B, y \in B
| c = Definition of $\sim$
}}
{{eqn | o = \leadstoandfrom
| r = \exists B \in \CC_x : y \in B
| c = equivalent definition
}}
{{eqn | o = \leadstoandfrom
| r = y \i... | Equivalence of Definitions of Component/Equivalence Class equals Union of Connected Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Component/Equivalence_Class_equals_Union_of_Connected_Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Component/Equivalence_Class_equals_Union_of_Connected_Sets | [
"Equivalence of Definitions of Component"
] | [
"Definition:Topological Space",
"Definition:Equivalence Relation",
"Definition:Connected Points (Topology)",
"Definition:Equivalence Class"
] | [] |
proofwiki-16094 | Equivalence of Definitions of Component/Union of Connected Sets is Maximal Connected Set | Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in T$.
Let $\CC_x = \set {A \subseteq S : x \in A \land A \text{ is connected in } T}$
Let $\ds C = \bigcup \CC_x$
Then $C$ is a maximal connected set of $T$. | Let $\tilde C$ be any connected set such that:
:$C \subseteq \tilde C$
Then $x \in \tilde C$.
Hence $\tilde C \in \CC_x$.
From Set is Subset of Union,
:$\tilde C \subseteq C$.
Hence $\tilde C = C$.
It follows that $C$ is a maximal connected set of $T$ by definition. | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in T$.
Let $\CC_x = \set {A \subseteq S : x \in A \land A \text{ is connected in } T}$
Let $\ds C = \bigcup \CC_x$
Then $C$ is a [[Definition:Maximal Set|maximal]] [[Definition:Connected Set (Topology)|connected set]] of $... | Let $\tilde C$ be any [[Definition:Connected Set (Topology)|connected set]] such that:
:$C \subseteq \tilde C$
Then $x \in \tilde C$.
Hence $\tilde C \in \CC_x$.
From [[Set is Subset of Union]],
:$\tilde C \subseteq C$.
Hence $\tilde C = C$.
It follows that $C$ is a [[Definition:Maximal Set|maximal]] [[Definition... | Equivalence of Definitions of Component/Union of Connected Sets is Maximal Connected Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Component/Union_of_Connected_Sets_is_Maximal_Connected_Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Component/Union_of_Connected_Sets_is_Maximal_Connected_Set | [
"Equivalence of Definitions of Component"
] | [
"Definition:Topological Space",
"Definition:Maximal/Set",
"Definition:Connected Set (Topology)"
] | [
"Definition:Connected Set (Topology)",
"Set is Subset of Union",
"Definition:Maximal/Set",
"Definition:Connected Set (Topology)"
] |
proofwiki-16095 | Equivalence of Definitions of Component/Maximal Connected Set is Union of Connected Sets | Let $\tilde C$ be a maximal connected set of $T$ that contains $x$. | By definition:
:$\tilde C \in \CC_x$
From Set is Subset of Union:
:$\tilde C \subseteq C$
By maximality of $\tilde C$:
:$\tilde C = C$ | Let $\tilde C$ be a [[Definition:Maximal Set|maximal]] [[Definition:Connected Set (Topology)|connected set]] of $T$ that contains $x$. | By definition:
:$\tilde C \in \CC_x$
From [[Set is Subset of Union]]:
:$\tilde C \subseteq C$
By [[Definition:Maximal Set|maximality]] of $\tilde C$:
:$\tilde C = C$ | Equivalence of Definitions of Component/Maximal Connected Set is Union of Connected Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Component/Maximal_Connected_Set_is_Union_of_Connected_Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Component/Maximal_Connected_Set_is_Union_of_Connected_Sets | [
"Equivalence of Definitions of Component"
] | [
"Definition:Maximal/Set",
"Definition:Connected Set (Topology)"
] | [
"Set is Subset of Union",
"Definition:Maximal/Set"
] |
proofwiki-16096 | Binomial Coefficient n Choose j in terms of n-2 Choose r | Let $n \in \Z$ such that $n \ge 4$.
Let $\dbinom n k$ denote a binomial coefficient for $k \in \Z$.
Then:
:$\dbinom n k = \dbinom {n - 2} {k - 2} + 2 \dbinom {n - 2} {k - 1} + \dbinom {n - 2} k$
for $2 \le k \le n - 2$. | {{begin-eqn}}
{{eqn | l = \dbinom n k
| r = \dbinom {n - 1} k + \dbinom {n - 1} {k - 1}
| c = Pascal's Rule
}}
{{eqn | r = \paren {\dbinom {n - 2} {k - 1} + \dbinom {n - 2} k} + \paren {\dbinom {n - 2} {k - 2} + \dbinom {n - 2} {k - 1} }
| c = Pascal's Rule (twice)
}}
{{eqn | r = \dbinom {n - 2} {k - ... | Let $n \in \Z$ such that $n \ge 4$.
Let $\dbinom n k$ denote a [[Definition:Binomial Coefficient|binomial coefficient]] for $k \in \Z$.
Then:
:$\dbinom n k = \dbinom {n - 2} {k - 2} + 2 \dbinom {n - 2} {k - 1} + \dbinom {n - 2} k$
for $2 \le k \le n - 2$. | {{begin-eqn}}
{{eqn | l = \dbinom n k
| r = \dbinom {n - 1} k + \dbinom {n - 1} {k - 1}
| c = [[Pascal's Rule]]
}}
{{eqn | r = \paren {\dbinom {n - 2} {k - 1} + \dbinom {n - 2} k} + \paren {\dbinom {n - 2} {k - 2} + \dbinom {n - 2} {k - 1} }
| c = [[Pascal's Rule]] (twice)
}}
{{eqn | r = \dbinom {n - ... | Binomial Coefficient n Choose j in terms of n-2 Choose r | https://proofwiki.org/wiki/Binomial_Coefficient_n_Choose_j_in_terms_of_n-2_Choose_r | https://proofwiki.org/wiki/Binomial_Coefficient_n_Choose_j_in_terms_of_n-2_Choose_r | [
"Binomial Coefficients"
] | [
"Definition:Binomial Coefficient"
] | [
"Pascal's Rule",
"Pascal's Rule",
"Definition:Negative/Integer"
] |
proofwiki-16097 | Sum of Sequence of Binomial Coefficients by Powers of 2 | {{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 0}^n 2^j \binom n j
| r = \dbinom n 0 + 2 \dbinom n 1 + 2^2 \dbinom n 2 + \dotsb + 2^n \dbinom n n
| c =
}}
{{eqn | r = 3^n
| c =
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = 3^n
| r = \paren {2 + 1}^n
}}
{{eqn | r = \sum_{j \mathop = 0}^n 2^j 1^{n - j} \binom n j
| c = Binomial Theorem
}}
{{eqn | r =\sum_{j \mathop = 0}^n 2^j \binom n j
}}
{{end-eqn}}
{{qed}} | {{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 0}^n 2^j \binom n j
| r = \dbinom n 0 + 2 \dbinom n 1 + 2^2 \dbinom n 2 + \dotsb + 2^n \dbinom n n
| c =
}}
{{eqn | r = 3^n
| c =
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = 3^n
| r = \paren {2 + 1}^n
}}
{{eqn | r = \sum_{j \mathop = 0}^n 2^j 1^{n - j} \binom n j
| c = [[Binomial Theorem]]
}}
{{eqn | r =\sum_{j \mathop = 0}^n 2^j \binom n j
}}
{{end-eqn}}
{{qed}} | Sum of Sequence of Binomial Coefficients by Powers of 2/Proof 1 | https://proofwiki.org/wiki/Sum_of_Sequence_of_Binomial_Coefficients_by_Powers_of_2 | https://proofwiki.org/wiki/Sum_of_Sequence_of_Binomial_Coefficients_by_Powers_of_2/Proof_1 | [
"Binomial Coefficients",
"Sum of Sequence of Binomial Coefficients by Powers of 2"
] | [] | [
"Binomial Theorem"
] |
proofwiki-16098 | Sum of Sequence of Binomial Coefficients by Powers of 2 | {{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 0}^n 2^j \binom n j
| r = \dbinom n 0 + 2 \dbinom n 1 + 2^2 \dbinom n 2 + \dotsb + 2^n \dbinom n n
| c =
}}
{{eqn | r = 3^n
| c =
}}
{{end-eqn}} | The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\ds \sum_{j \mathop = 0}^n 2^j \binom n j = 3^n$
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 0}^0 2^j \binom n j
| r = \dbinom 0 0
| c =
}}
{{eqn | r = 1
| c =
}}
{{eqn | r = 3^0... | {{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 0}^n 2^j \binom n j
| r = \dbinom n 0 + 2 \dbinom n 1 + 2^2 \dbinom n 2 + \dotsb + 2^n \dbinom n n
| c =
}}
{{eqn | r = 3^n
| c =
}}
{{end-eqn}} | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \sum_{j \mathop = 0}^n 2^j \binom n j = 3^n$
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 0}^0 2^j \binom n j
| r = \d... | Sum of Sequence of Binomial Coefficients by Powers of 2/Proof 2 | https://proofwiki.org/wiki/Sum_of_Sequence_of_Binomial_Coefficients_by_Powers_of_2 | https://proofwiki.org/wiki/Sum_of_Sequence_of_Binomial_Coefficients_by_Powers_of_2/Proof_2 | [
"Binomial Coefficients",
"Sum of Sequence of Binomial Coefficients by Powers of 2"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Binomial Coefficient with Zero",
"Binomial Coefficient with Self",
"Pascal's Rule",
"Translation of Index Variable of Summation",
... |
proofwiki-16099 | Sum of Sequence of n Choose 2 | Let $n \in \Z$ be an integer such that $n \ge 2$.
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 2}^n \dbinom j 2
| r = \dbinom 2 2 + \dbinom 3 2 + \dbinom 4 2 + \dotsb + \dbinom n 2
| c =
}}
{{eqn | r = \dbinom {n + 1} 3
| c =
}}
{{end-eqn}}
where $\dbinom n j$ denotes a binomial coefficient. | We can rewrite the {{LHS}} as:
:$\ds \sum_{j \mathop = 0}^m \dbinom {2 + j} 2$
where $m = n - 2$.
From Rising Sum of Binomial Coefficients:
:$\ds \sum_{j \mathop = 0}^m \binom {n + j} n = \binom {n + m + 1} {n + 1}$
The result follows by setting $n = 2$ and changing the upper index from $m$ to $n - 2$.
{{qed}} | Let $n \in \Z$ be an [[Definition:Integer|integer]] such that $n \ge 2$.
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 2}^n \dbinom j 2
| r = \dbinom 2 2 + \dbinom 3 2 + \dbinom 4 2 + \dotsb + \dbinom n 2
| c =
}}
{{eqn | r = \dbinom {n + 1} 3
| c =
}}
{{end-eqn}}
where $\dbinom n j$ denotes a [[Def... | We can rewrite the {{LHS}} as:
:$\ds \sum_{j \mathop = 0}^m \dbinom {2 + j} 2$
where $m = n - 2$.
From [[Rising Sum of Binomial Coefficients]]:
:$\ds \sum_{j \mathop = 0}^m \binom {n + j} n = \binom {n + m + 1} {n + 1}$
The result follows by setting $n = 2$ and changing the upper index from $m$ to $n - 2$.
{{qed}... | Sum of Sequence of n Choose 2/Proof 1 | https://proofwiki.org/wiki/Sum_of_Sequence_of_n_Choose_2 | https://proofwiki.org/wiki/Sum_of_Sequence_of_n_Choose_2/Proof_1 | [
"Binomial Coefficients",
"Sum of Sequence of n Choose 2"
] | [
"Definition:Integer",
"Definition:Binomial Coefficient"
] | [
"Rising Sum of Binomial Coefficients"
] |
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