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statistics | probability of hand having cards from exactly two hands | https://math.stackexchange.com/questions/1447262/probability-of-hand-having-cards-from-exactly-two-hands | <p>Can you guys answer this question regarding part (c)? I am confused of what he means.</p>
<p>A deck of <span class="math-container">$52$</span> cards is mixed well, and <span class="math-container">$5$</span> cards are dealt.
(a) It can be shown that (disregarding the order in which the cards are dealt) there are <s... | <p>By symmetry, there are also $63206$ hands with diamonds and hearts, with both suits represented. </p>
<p>There are also $63206$ hands with diamonds and clubs, with both suits represented.</p>
<p>The same applies to spades and hearts, to spades and clubs, and so on. Make a list. There are exactly $6$ types of hands... | 500 |
statistics | Extended sample; mode, median, mean | https://math.stackexchange.com/questions/1484493/extended-sample-mode-median-mean | <p>Given a sample (the scope is 72 elements) with mode=54 mean=55,7 median=54,5. The 73th value of the extended sample is 56. What can I say about the mode, median and mean of the extended sample?</p>
<p>Well, the updated mean is easy to calculate. </p>
<p>To make a statement about the median I know the 36th and the ... | <p>The mean of $n = 72$ observations is
$$\bar X_{72} = \frac{\sum_{i=1}^{72} X_i}{72} = 55.7.$$
You can use this information to find $\sum_{i=1}^{72} X_i$ and
from there to find $\sum_{i=1}^{73} X_i$ and from there
$\bar X_{73}.$</p>
<p>To find the median of all 73 observations you need to know
observation number $7... | 501 |
statistics | Finding MLE of a distribution density, and derive a new MLE based off of the parameter $\theta$ | https://math.stackexchange.com/questions/2979184/finding-mle-of-a-distribution-density-and-derive-a-new-mle-based-off-of-the-par | <p>Given a distribution with density
<span class="math-container">$$f(x)=\frac{x}{\theta^2}\exp(\frac{-x}{\theta})$$</span>
How do I find the Maximum Likelihood Estimator of <span class="math-container">$\log(θ +7)$</span> ?</p>
<p>I have found the MLE of <span class="math-container">$\theta$</span> as
<span class="... | <p>Once you have the MLE <span class="math-container">$\hat{\theta}$</span> of <span class="math-container">$\theta$</span>, the MLE of <span class="math-container">$f(\theta)$</span> is <span class="math-container">$f(\hat{\theta})$</span>, since in both cases we're finding the point in parameter space that maximises ... | 502 |
statistics | What's the difference between Gamma(alpha, lambda) and Gamma(alpha, beta)? | https://math.stackexchange.com/questions/2979196/whats-the-difference-between-gammaalpha-lambda-and-gammaalpha-beta | <p>I've found different formulas for the Gamma distribution, one where Gamma(alpha, lambda) has an expected value of alpha/lambda due to the Gamma distribution turning into the following image: <a href="https://prnt.sc/lcq5zq" rel="nofollow noreferrer">https://prnt.sc/lcq5zq</a>. However, in other sites I see Beta bein... | <p>Both forms are correct as they are just two different parametrizations for the <a href="https://en.m.wikipedia.org/wiki/Gamma_distribution" rel="nofollow noreferrer">Gamma Distribution.</a> The parameter <span class="math-container">$\lambda$</span> wouldn't mean the same thing as the parameter <span class="math-con... | 503 |
statistics | Positive parameters for Gamma random variables | https://math.stackexchange.com/questions/4170608/positive-parameters-for-gamma-random-variables | <p>I am sorry for the poor quality of this question: For <span class="math-container">$\Gamma(\alpha,\beta)$</span> random variables, why do we assume <span class="math-container">$\alpha>0$</span> and <span class="math-container">$\beta>0$</span>?</p>
| <p>The gamma density is the following, for <span class="math-container">$x>0$</span></p>
<p><span class="math-container">$$f_X(x,a,b)=\frac{b^a}{\Gamma(a)}x^{a-1}e^{-bx}$$</span></p>
<p>it is easy to prove that its integral cannot converge if <span class="math-container">$a,b$</span> are not both positive</p>
| 504 |
statistics | Basic Math Question I think... | https://math.stackexchange.com/questions/4176527/basic-math-question-i-think | <p>I apologize for the title. I am not even sure how to phrase this question per se. I feel like this should be easy and yet I am questioning my thinking. Here's the scenario:</p>
<p>I have two groups. Group A has 50 members. Group B has 400 members. What I want to know is what the calculation would be to adjust ... | <p>Comment continued with results from test procedures in R:</p>
<pre><code>prop.test(c(5,20),c(50,400), cor=F)
2-sample test for equality of proportions
without continuity correction
data: c(5, 20) out of c(50, 400)
X-squared = 2.1176, df = 1, p-value = 0.1456
alternative hypothesis: two.sided
95 p... | 505 |
statistics | which measure of central tendency might be used by the boss who is against a pay rise for other employees? | https://math.stackexchange.com/questions/2993373/which-measure-of-central-tendency-might-be-used-by-the-boss-who-is-against-a-pay | <p>In an office of 20 people ther are only 4 salary levels paid :
50 000 (1 person), 42 000 (3 people), 35 000 (6 people), 28 000 (10 people).</p>
<p>I calculate the mean = 33300, the median = 30000 (usd).</p>
<p>But I am not sure which measure of central tendency might be used by the boss who is against a pay rise f... | <p>The median of <span class="math-container">$30,000$</span> is more convincing because the people who are making <span class="math-container">$28000$</span> are only <span class="math-container">$2,000$</span> below the median while the rest of them are making much higher than the median.</p>
<p>Thus more people fee... | 506 |
statistics | Statistics - Regression | https://math.stackexchange.com/questions/1178005/statistics-regression | <p>A sample of $40$ women is obtained and their heights in inches and pulse rate in beats per minute are measured. The linear correlation coefficient is $0.221$ and the equation of the regression line is $y = 18.5+0.860x$ where $x$ represents height and $y$ the pulse rate. The mean of the $40$ heights is $62.8$ inches ... | <p>This is not really an answer because I think something is wrong with the
information given. Perhaps you can make some sense of the approach anyway.</p>
<p>Regression line: y = 18.5 + 0.86x. If x = 65, then the predicted height is
18.5 + 0.86(65) = 74.4, which is close but not exactly the given answer 73.2.</p>
<p... | 507 |
statistics | Determining probability after standardizing for given problem | https://math.stackexchange.com/questions/4183246/determining-probability-after-standardizing-for-given-problem | <p>During the log phase of bacterial growth the size of the colony grows exponentially. Let <span class="math-container">$R$</span> be the ratio of the biomass at time 1 hour to the initial biomass. Then <span class="math-container">$R=e^r$</span> where <span class="math-container">$r$</span> is the instantaneous growt... | 508 | |
statistics | How many possibilities can a 10x5 grid with somewhat even distribution produce? | https://math.stackexchange.com/questions/3153176/how-many-possibilities-can-a-10x5-grid-with-somewhat-even-distribution-produce | <p>Imagine a 10 x 5 grid where each square can be either 1 or 0. However, each row (10 squares) must contain five 1's and five 0's. Therefore, each grid (of 50 squares) has twenty five 1's and twenty five 0's but their distribution is somewhat controlled in that each row must contain five of each.</p>
<p>How many poss... | <p>For each row, choose five positions to have 1's. The remaining 5 positions have zeros.
Do this for each row.</p>
<p>That is:</p>
<p><span class="math-container">$$\dbinom{10}{5}^5 = 252^5$$</span></p>
| 509 |
statistics | How to forecast revenue with my data? | https://math.stackexchange.com/questions/4186587/how-to-forecast-revenue-with-my-data | <p>I work for a law firm and I want to know whether I can accurately predict our future revenue based on data I've pulled from our SQL Server database. Here is the information that I know based on some queries:</p>
<ol>
<li><p>It takes on average, 333 days from the time we sign up a client until we get a settlement and... | 510 | |
statistics | Pearson correlation over subsamples compared to the whole sample | https://math.stackexchange.com/questions/4178919/pearson-correlation-over-subsamples-compared-to-the-whole-sample | <p>I work on 'real-life data' and to simplify, I have a sample of 10.000 observations of two variables <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> with a categorical variable <span class="math-container">$C$</span> that is either <span class="math-container">$C=0$</span> or <span... | 511 | |
statistics | Event uptime over a course of time | https://math.stackexchange.com/questions/4187831/event-uptime-over-a-course-of-time | <pre><code>Event Trigger Check Interval: 3 minutes
Trigger Chance of Event: 5%
Event Length: 10 minutes
</code></pre>
<p>The above are the 3 variables in this problem. To give it a story, let's say I click a button every <strong>3 minutes</strong>, with the first click at minute 3 (not 0). On click, there is a <stron... | <p>If I understand correctly, of the <span class="math-container">$(180)$</span> minutes to be considered, you want <span class="math-container">$\sum_{i=1}^{180} f(i)$</span>, where <span class="math-container">$f(i)$</span> is the probability of rain during the interval from the start of minute <span class="math-cont... | 512 |
statistics | Consider the mean combined SAT score for high school seniors is 1500, | https://math.stackexchange.com/questions/1883202/consider-the-mean-combined-sat-score-for-high-school-seniors-is-1500 | <p>"Consider the mean combined SAT score for high school seniors is 1500, and the standard deviation is 250. Calculate the percentage of students who scored at the following levels"...</p>
<p>Can anyone figure out what this question is asking for? I can find a Z-score then a corresponding probability but I don't think... | <p>It's difficult to guess from this fragment. If this is at the beginning of
using normal distributions in a basic probability or statistics course, my
guess is they want you to standardize and use printed normal tables (or
possibly software).</p>
<p>Assuming that the population is normal with mean $\mu = 1500$ and S... | 513 |
statistics | What if the relative error is undefined? | https://math.stackexchange.com/questions/2502932/what-if-the-relative-error-is-undefined | <p>The relative error is defined by the simple formula:</p>
<p>$$\text{Rel. Error} = \frac{|v_\text{approx}-v_\text{analytical}|}{v_\text{analytical}}$$</p>
<p>but what if the theoretical value $v_\text{analytical}$ should be $0$? then our relative error is undefined.... this is also quite a common occurs. If our ana... | <p>You could argue that in that particular case, the relative error is not a good measure.</p>
<p>Note that usually the relative error is defined a the ratio of the absolute error and the <em>absolute</em> true value, i.e.
$$
\mathrm{Rel. Error} = \frac{ |v_{\mathrm{approx}} - v_{\mathrm{analytical}}| }{ |v_\mathrm{an... | 514 |
statistics | When creating histograms, what is wrong with data values falling on boundaries? | https://math.stackexchange.com/questions/3211937/when-creating-histograms-what-is-wrong-with-data-values-falling-on-boundaries | <p>My textbook mentions this:</p>
<p>"To construct a histogram, first decide how many bars or intervals, also called classes, represent the data. Many
histograms consist of five to 15 bars or classes for clarity. The number of bars needs to be chosen. Choose a starting point
for the first interval to be less than the ... | <p>The intervals in a histogram is a range.</p>
<p>So if I made a histogram with bars <span class="math-container">$0-10$</span>, <span class="math-container">$10-20$</span>, <span class="math-container">$20-30$</span>, etc. </p>
<p>And a data point said <span class="math-container">$10$</span>, what would do? Would ... | 515 |
statistics | Proof that loss function for linear regression is an ellipsoid | https://math.stackexchange.com/questions/2909077/proof-that-loss-function-for-linear-regression-is-an-ellipsoid | <p>I cannot seem to find a proof that $f(\mathbf{\beta}) = \left\lVert \mathbf{y}-\mathbf{X} \mathbf{\beta} \right\rVert^2$ is an ellipsoid, centered at the OLS solution $\hat{\beta}$. Can anyone show how to convert it to the quadratic form of a general ellipsoid, i.e. $(\beta - \hat{\beta})^T \mathbf{A} (\beta - \hat{... | <p>The function $(\beta - \hat{\beta})^\top A (\beta - \hat{\beta})$ is zero when $\beta = \hat{\beta}$, yet $f(\hat{\beta}) = \|y - X \hat{\beta}\|^2$ might not be zero.</p>
<p>The form you are looking for is $(\beta - \hat{\beta})^\top A (\beta - \hat{\beta}) + c$ for some vector $c$ (which will turn out to be $\|y ... | 516 |
statistics | If $E(U|X)=0$, then $E(U)=0$? | https://math.stackexchange.com/questions/2924578/if-eux-0-then-eu-0 | <blockquote>
<p>If <span class="math-container">$U$</span> and <span class="math-container">$X$</span> are random variables such that <span class="math-container">$E(U|X)=0$</span>, then <span class="math-container">$E(U)=0$</span>.</p>
</blockquote>
<p>Really? how to prove?</p>
| <p><span class="math-container">$E(U\mid X)$</span> is actually a random variable which is a certain function of <span class="math-container">$X$</span> (say, <span class="math-container">$g(X)$</span> for instance). There's a property that says that if you take its expectation you get
<span class="math-container">$$E\... | 517 |
statistics | Expectation of mle for exponential r.v.'s with censoring | https://math.stackexchange.com/questions/2928389/expectation-of-mle-for-exponential-r-v-s-with-censoring | <p>Let <span class="math-container">$X_1, X_2, \ldots, X_n$</span> be i.i.d exponential random variables with parameter <span class="math-container">$\lambda$</span>, where the form of the distribution for each <span class="math-container">$x_i$</span> is</p>
<p><span class="math-container">$$f(x_i) =\lambda e^{-\lamb... | 518 | |
statistics | Is there any relation between a independece of a event and a mutual exclusive event? | https://math.stackexchange.com/questions/2941568/is-there-any-relation-between-a-independece-of-a-event-and-a-mutual-exclusive-ev | <p>This questions was proposed by our statistics teacher as home work.</p>
<p>I have been looking for any reasonable explanation but so far I am getting more and more confused differentiating between the two concepts.</p>
<p>Could someone please clarify them for me?</p>
| <p>If A and B are independent <span class="math-container">$$P(A \cap B)=P(A)P(B)$$</span></p>
<p>If they are mutually exclusive then <span class="math-container">$$P(A \cap B)=0$$</span></p>
<p>So the only way that two outcomes can be both independent and mutually exclusive is if at least one of the outcomes occurs ... | 519 |
statistics | How to calculate statistical "difference" of two samples in 0-100 range? | https://math.stackexchange.com/questions/2971770/how-to-calculate-statistical-difference-of-two-samples-in-0-100-range | <p>There are two groups of people, target and neutral</p>
<p>There are two group of events X and others</p>
<p>We're making assumption that people from target group react on event X very different to others.</p>
<p>We have some sets of reaction numeric values on different type of events by those two groups. And afte... | <p>If the data are approximately normal then use a <a href="https://en.wikipedia.org/wiki/Welch%27s_t-test" rel="nofollow noreferrer">Welch 2-sample t test</a>. This test does not assume
that the two groups are sampled from populations with equal <em>variances.</em></p>
<p>When you mention the <code>0-100</code> range... | 520 |
statistics | Minimum size of test given that power function is at least 0.6 in p = 0.3 | https://math.stackexchange.com/questions/2983953/minimum-size-of-test-given-that-power-function-is-at-least-0-6-in-p-0-3 | <p>The question is as follows:</p>
<p>Let X be a variable with bin(25,p)-distribution. We test <span class="math-container">$H_{0}$</span>: p <span class="math-container">$\geq$</span> 0.4 against <span class="math-container">$H_{1}$</span>: p < 0.4. If we want a power function of at least 0.6 in p = 0.3, how large... | <p>Out of the four quantities significance level <span class="math-container">$\alpha,$</span>
power, sample size, and difference <span class="math-container">$\Delta$</span> detected,
a power computation typically specifies three and finds
the remaining one. </p>
<p>The following power curve from Minitab uses <span c... | 521 |
statistics | How to prove that $\sum_{i<j}(X_i-X_j)^2=n\sum_{i=1}^{n}(X_i-\bar{X})^2$ | https://math.stackexchange.com/questions/3045070/how-to-prove-that-sum-ijx-i-x-j2-n-sum-i-1nx-i-barx2 | <p>In a example about U-statistics, <span class="math-container">$h(x_1,x_2)=\frac 12(x_1-x_2)^2$</span>, then
<span class="math-container">$$U_n=\frac{2}{n(n-1)}\sum_{i<j}\frac{(X_i-X_j)^2}{2}=\frac{1}{n-1}\sum_{i=1}^{n}(X_i-\bar{X})^2$$</span>
I don't know how to prove it completely.</p>
| <p>We know that (I found it <a href="https://math.stackexchange.com/questions/1187687/expansion-of-the-square-of-the-sum-of-n-numbers">here</a>)
<span class="math-container">\begin{equation}
\left( \sum_{n=1}^N a_n \right)^2 = \sum_{n=1}^N a_n^2 + 2 \sum_{j=1}^{N}\sum_{i=1}^{j-1} a_i a_j
\end{equation}</span>
So using... | 522 |
statistics | Easy hypothesis testing in discrete case (uniform distribution) | https://math.stackexchange.com/questions/3072496/easy-hypothesis-testing-in-discrete-case-uniform-distribution | <p>We have an estimator <span class="math-container">$\hat{X}$</span> of <span class="math-container">$N$</span> which takes values in <span class="math-container">$\{1,2,\cdots,N\}$</span> with the following mass function:
<span class="math-container">$$P_N(\hat{X}=k) = \left(\frac{k}{N}\right)^n-\left(\frac{k-1}{N}\r... | <p>I suspect you mean to say you would reject <span class="math-container">$H_0$</span> when <span class="math-container">$\hat{X}=22$</span>. You would also reject <span class="math-container">$H_0$</span> when <span class="math-container">$\hat{X}=21$</span> since that too is inconsistent with <span class="math-cont... | 523 |
statistics | Prove that statistics is sufficient | https://math.stackexchange.com/questions/3085755/prove-that-statistics-is-sufficient | <p>in my wordbook it is said to be true, but I would like to know how to prove it. Let <span class="math-container">$X_1,X_2,...,X_n$</span> be distributed <span class="math-container">$Exp(\lambda)$</span> and <span class="math-container">$T(X_1,X_2,...,X_n)=X_1+X_2+...+X_n$</span> Prove that statistics T is sufficien... | <p>Hint:</p>
<p>Calculate the mutual density</p>
<p><span class="math-container">$$f_\lambda(X_1, \dots, X_n) = \prod_{i=1}^n f_\lambda (X_i)$$</span> and the sum you are looking for will apear. </p>
| 524 |
statistics | Probability on a time basis | https://math.stackexchange.com/questions/3168750/probability-on-a-time-basis | <p>They were talking about volcanoes today on TV, specifically the Popocatépetl and the threat of pyroclastic flows if a Plinian eruption happened. It was mentioned that because an eruption didn't occur in 1000 years (AD 800) there is an increasing probability.
Popular science, so intuitive, but what can be calculated ... | 525 | |
statistics | Why do we associate discrete values to data from a continuous distribution? | https://math.stackexchange.com/questions/3174171/why-do-we-associate-discrete-values-to-data-from-a-continuous-distribution | <p>If we think about a random variable following a discrete distribution then I can accept the association of a discrete value in data to the variable. For example, let's talk about n independent basketball shots each with probability p of making it into the hoop. The number of shots that make it in is distributed <spa... | <p>Here is a distinction between discrete and continuous random variables
that I think you may find useful.</p>
<p><strong>Discrete random variable:</strong> Suppose a random variable can take integer values <span class="math-container">$0, 1, 2, \dots, 10.$</span> A random variable <span class="math-container">$X \si... | 526 |
statistics | when to use t-test or z-test when computing p-value | https://math.stackexchange.com/questions/3185558/when-to-use-t-test-or-z-test-when-computing-p-value | <p>Here are two seperate question</p>
<p>Q1) An observed sample of four observations from a <span class="math-container">$N(u, \sigma^2)$</span> distribution has mean <span class="math-container">$62.75$</span> and standard deviation <span class="math-container">$4.57$</span>. Assess the hypothesis <span class="math-c... | <p>Some confusion here. Perhaps printouts from software will help
you get oriented.</p>
<p><strong>1) Use t test--with sample standard deviation from data:</strong> In Minitab, the <em>two-sided</em> t test (alternative <span class="math-container">$H_a:\mu \ne 66)$</span>
results in the following output:</p>
<pre><... | 527 |
statistics | What is this formula for sample standard deviation? | https://math.stackexchange.com/questions/3214115/what-is-this-formula-for-sample-standard-deviation | <p>My textbook mentioned this formula without any explanation: </p>
<p>s<sub>x</sub> = <span class="math-container">$\sqrt{\frac{{∑fm}^2}{n} − \bar{x}^2}$</span>
where
s<sub>x</sub> = sample standard deviation
<span class="math-container">$\bar{x}$</span> = sample mean</p>
<p>I thought the sample standard deviation ... | 528 | |
statistics | How to find joint PDF of Z and Y where Z = X + Y | https://math.stackexchange.com/questions/3216408/how-to-find-joint-pdf-of-z-and-y-where-z-x-y | <p>Let there be two random variables X and Y. A third random variable Z is defined as Z = X + Y. Suppose we are given <span class="math-container">$f_{XY}$</span> (the joint probability density function of X and Y).</p>
<p>How do we calculate <span class="math-container">$f_{ZY}$</span> (the joint probability density ... | <p>A starndard idea in this type of exercise is to start by fiding the cdf of <span class="math-container">$(Z,Y)$</span>:
<span class="math-container">\begin{align*}
P(Z \leq z_0, Y \leq y_0)
&= P(X + Y \leq z_0, Y \leq y_0) \\
&= \int_{-\infty}^{y_0}\int_{-\infty}^{z_0-y}f_{X,Y}(x,y)dxdy
\end{align*}</span... | 529 |
statistics | How to statistically compare the outcome of two classification algorithms? | https://math.stackexchange.com/questions/3272177/how-to-statistically-compare-the-outcome-of-two-classification-algorithms | <p>I have two deep learning classifiers and I want to make a test statistic to compare their performance against each other. The two algorithms have been tested on the same 60 samples and have an accuracy of 78.6% and 91.8% respectively. How do I compare their performance? Would I use Students paired t-test in this sce... | 530 | |
statistics | Pascal's triangle and inheritance laws | https://math.stackexchange.com/questions/3363570/pascals-triangle-and-inheritance-laws | <p>I'm trying to get a grasp on genetic inheritance laws. Assuming that chromosomes remain intact, would the 23rd row of Pascal's triangle accurately show the distribution of inheritance from a set of grandparents? The first number would be the occurrences of the 23 chromosomes split 0-23,then 1-22, 2-21 ...22-1, 23-... | <p>I would say that under normal circumstances, the mother contributes one chromosome in each of the pairs <span class="math-container">$1$</span> to <span class="math-container">$23$</span> and the father contributes the other.
Among the <span class="math-container">$23$</span> chromosomes contributed by the mother, e... | 531 |
statistics | standard deviation of two samples | https://math.stackexchange.com/questions/3424318/standard-deviation-of-two-samples | <p>there are two samples, both with standard deviation 8.5.
If one combines the two samples together, find the relation between the standard deviation and 17.</p>
<p>I figure it cannot be determined. But I am not sure how to argue rigorously. Thank you!</p>
| 532 | |
statistics | Expected value of $x^TAx$ | https://math.stackexchange.com/questions/3459410/expected-value-of-xtax | <p>I have a question on this specific question from the past entrance examination of a university.</p>
<p><a href="https://www.ism.ac.jp/senkou/kakomon/math_20190820.pdf" rel="nofollow noreferrer">https://www.ism.ac.jp/senkou/kakomon/math_20190820.pdf</a></p>
<p>I assume that the mean vector of the d-dimensional vect... | <p><span class="math-container">$x^TAx=\sum_{i=1}^d \sum_{j=1}^d a_{ij}x_ix_j$</span> where <span class="math-container">$a_{ij}$</span> are elements of <span class="math-container">$A$</span>. </p>
<p>Then use linearity of expectation and given variance-covariance matrix.</p>
| 533 |
statistics | When do you have to apply continuity correction? | https://math.stackexchange.com/questions/3465210/when-do-you-have-to-apply-continuity-correction | <p>I've solved the following problem. </p>
<blockquote>
<p>An electronics firm receives, on the average, fifty orders per week for a particular silicon chip. If the company has sixty chips on hand, use the Central Limit Theorem to approximate the probability that they will be unable to fill all their orders for the ... | <p>It's usually better to do the continuity correction when approximating an integer-valued distribution by a normal distribution, but it may not make any significant difference when the normal approximation is very accurate, which occurs for Poisson(<span class="math-container">$\lambda$</span>) for very large <span c... | 534 |
statistics | Measuring Point Density in $\mathbb{R}^2$ | https://math.stackexchange.com/questions/3510381/measuring-point-density-in-mathbbr2 | <p>Suppose we are given a non-empty set of two or more points in <span class="math-container">$\mathbb{R}^2$</span>, <span class="math-container">$P$</span>, and we would like to quantify how <em>dense</em> those points are to one another. Let <span class="math-container">$\mathcal{D}(P)$</span> represent the density o... | <p>If <span class="math-container">$P$</span> is contained a known region, then one possible metric of density is to measure how much area the points in <span class="math-container">$P$</span> cover compared to the total region, which can be computed as the ratio of the area of the convex hull to the area of the region... | 535 |
statistics | Standard Error and Standard Deviations | https://math.stackexchange.com/questions/3552581/standard-error-and-standard-deviations | <p>How it s possible to calculate standard error for single sample, While standard error is defined as variance in different sample means?</p>
| <p>The population has a (presumably unknown) variance. If you take a sample of size <span class="math-container">$n$</span> with replacement then the variance of the sample mean is <span class="math-container">$\frac1n$</span> times the population variance. Taking the square root gives the standard deviation.</p>
<p... | 536 |
statistics | What is the UMVUE of $\exp(-2\lambda)$ for X is a random variable with poisson distribution~poisson $(\lambda)$ | https://math.stackexchange.com/questions/3564547/what-is-the-umvue-of-exp-2-lambda-for-x-is-a-random-variable-with-poisson-d | <p>If <span class="math-container">$X_1,X_2,…,X_n\sim Pois(λ)$</span>, find the UMVUE of <span class="math-container">$\exp(−2λ)$</span>.
I know based on lehmann scheffe theorem, the step is</p>
<p>(1)find <span class="math-container">$q(x)$</span> an unbiased estimater of <span class="math-container">$\exp(−2λ)$</spa... | <p>An easier approach is to take the approach in the two linked questions, using an initial unbiased estimator of <span class="math-container">$\exp(−2\lambda)$</span> to be <span class="math-container">$1$</span> if <span class="math-container">$X_1+X_2=0$</span> and to be <span class="math-container">$0$</span> if <s... | 537 |
statistics | mean and standard derivation about battery | https://math.stackexchange.com/questions/3609189/mean-and-standard-derivation-about-battery | <p>The batteries produced in a factory are tested before packing: 1.5% of batteries are found to be faulty, and are scrapped. Whether or not a battery is faulty is independent of each other. Experience suggested that a ‘good’ battery could last for 36 to 45 hours when used, and that all times within this range are equa... | <p><strong>Background.</strong> I suppose the <span class="math-container">$n = 100$</span> batteries are used sequentially, so that
the total time the box lasts is <span class="math-container">$T = X_1 + X_2 + \cdots + X_{100},$</span>
where <span class="math-container">$X_i \sim \mathsf{Unif}(35.45).$</span> </p>
<... | 538 |
statistics | $k$ being given, probability that when selecting two numbers $a,b$ in $\{1,2,\cdots n\}$, one has $a<k<b$ | https://math.stackexchange.com/questions/3624952/k-being-given-probability-that-when-selecting-two-numbers-a-b-in-1-2-cd | <p>Two numbers are choosen at random from the sequence of number 1,2,3,.........n. what is the probability that one of them is less than k and other is greater than k?</p>
| <p>If <span class="math-container">$k=1$</span> and <span class="math-container">$k=n$</span>, the probability is <span class="math-container">$0$</span>.</p>
<p>Let us assume that <span class="math-container">$1<k<n$</span>.</p>
<p>First of all, the total number of choices is <span class="math-container">$B=\b... | 539 |
statistics | Ratio of Certain Dependent Random Variables | https://math.stackexchange.com/questions/2758870/ratio-of-certain-dependent-random-variables | <p><strong>Disclaimer</strong>: This problem is for my own understanding and not for a class in any way. </p>
<p>Greetings!</p>
<p>I am trying to solve the following problem but I am unsure how to proceed beyond what I have below. The trouble I am having is that $Z$ and $W$ and not independent of one another and so I... | <p><strong>Comment:</strong> To me, this is still not clearly stated. (In particular, I don't follow your last two displayed formulas.) Here is a speculative interpretation
of a special case. First, you select $W \sim \mathsf{Unif}(0, 5).$ Then
you select $Z \sim \mathsf{Unif}(W, 10).$ So $Z$ has support $(0,10),$
but ... | 540 |
statistics | Identification of the experimental design | https://math.stackexchange.com/questions/2793969/identification-of-the-experimental-design | <p>We have the following structure:</p>
<p>$\begin{bmatrix}
A & B & C & D\\
A & C & B & D\\
B & A & C & C\\
A & A & B & C
\end{bmatrix}$</p>
<p>It is required to tell which design is this CRD,LSD,RBD or factorial design?</p>
<p>I know it can't be LSD as it has same ... | 541 | |
statistics | Probability using statistics | https://math.stackexchange.com/questions/2750166/probability-using-statistics | <p>Scenario: A company that makes cartons finds the probability of producing a carton with a puncture is 0.05. The probability that a carton has a smashed corner is 0.08. The probability that a carton has a puncture and a smashed corner is 0.004. </p>
<p>Question: If a quality inspector randomly selects a carton, find... | <p>Let $P$ and $S$ respectively be the events "the carton has a puncture" and "the carton has a smashed corner":</p>
<p>$$P(P \cup S) = P(P) + P(S) - P(P \cap S)$$</p>
<p>Thus, what you are looking for is 0.126</p>
<p>The property invoked is true for all <a href="https://en.wikipedia.org/wiki/Probability#Not_mutuall... | 542 |
statistics | How is standard deviation different from average deviation? | https://math.stackexchange.com/questions/2795144/how-is-standard-deviation-different-from-average-deviation | <p>Standard deviation is used as a measure of deviation or distribution in a sample or a population.Similarly mean is used as a measure of something in a group(say average marks of a group of students).How is standard deviation different from average deviation?</p>
| <p>Let $(x_i)$ be a data set. Denote by the average deviation $D$. Then</p>
<p>$$ D = \frac{1}{n} \sum_{i=1}^n (x_i - \mu) $$</p>
<p>and</p>
<p>$$ \sigma = \sqrt{ \frac{1}{n} \sum_{i = 1}^n (x_i - \mu) ^2 } $$</p>
<p>Certainly two different quantities. Variance (the square of standard deviation) measure the average... | 543 |
statistics | Is probability mass function (pmf) unique? | https://math.stackexchange.com/questions/2795856/is-probability-mass-function-pmf-unique | <p>It is known that probability density function (pdf) is not unique, but can we say the same about pmf?</p>
<p>Also, what can be the possible example where pdf or pmf may not exist?</p>
| <p>If you want a simple example where it may not exist, consider the discrete uniform on -inf to inf. That is each event on the Reals is equally likely. </p>
| 544 |
statistics | Statistic Problem | https://math.stackexchange.com/questions/2776897/statistic-problem | <p>An urn contains 3 red balls and 2 blue balls. A ball is drawn.
If the ball is red, it is kept out of the urn and a second ball is drawn from the urn.
If the ball is blue, then it is put back in the urn and a red ball is added to the urn.
Then a second ball is drawn from the urn.
(a) What is the probability that both... | <p>a. $3/5\cdot 2/4 = 3/10$</p>
<p>b. The probability of RR is $3/5\cdot 2/4 = 3/10$</p>
<p>The probability of BR is $2/5\cdot 4/6 = 4/15$</p>
<p>The probability of the first being B given the second is R is.....</p>
<p>$$P(A|B) = \frac{P(B|A)\cdot P(A)}{P(B)} = \frac{4/6\cdot 2/5}{(4/6\cdot 2/5) + (2/4\cdot 3/5)} ... | 545 |
statistics | Probability of event occurring based on dependent events | https://math.stackexchange.com/questions/2805564/probability-of-event-occurring-based-on-dependent-events | <p>I have a question which I'm not sure how to phrase hence i couldn't find similar questions:</p>
<p>Usually all the examples i find online refer to dependent events and their probability of happening based on the event they depend on.</p>
<p>But in my question the issue is finding the probability of that event they... | <p>To elaborate on the comments:</p>
<p>First of all, to proceed we need to assume something about how the events depend on each other. It can't be the case that they are strictly independent of each other. For example: seeing the aliens is strong evidence for the absence of rain, and as the absence of rain also in... | 546 |
statistics | Central limit theorem independent distributions | https://math.stackexchange.com/questions/2762246/central-limit-theorem-independent-distributions | <p>I'm confused by the statement of the central limit theorem we've been given:</p>
<p>'If you take the sum $X$ of $N$ independent variables $x_i$, each taken from a distribution with mean $\mu_i$, then the distribution for $X$ has the average $\Sigma\mu_i$.'</p>
<p>So does this mean that if I take one number $x_i$ f... | 547 | |
statistics | Methodology of Choosing Random Data | https://math.stackexchange.com/questions/2797391/methodology-of-choosing-random-data | <p>For a project, I have to randomly choose exoplanets out of a database, but since the database has different sections, I have to choose one section. If one section has more exoplanets than another and I am randomly choosing five exoplanets, will the five exoplanets be more diverse if I choose the bigger or smaller se... | 548 | |
statistics | What sample size is needed at the 95% confidence level, where the error (E) is 3 and the standard deviation is 20? | https://math.stackexchange.com/questions/2825564/what-sample-size-is-needed-at-the-95-confidence-level-where-the-error-e-is-3 | <p>What sample size is needed at the 95% confidence level, where the error (E) is 3 and the standard deviation is 20?</p>
<p>I can't figure out this question for the life of me and I am not sure of what formula to use. I know the answer is 171 but I would like to understand how we got there.</p>
| <p>$n = (\frac{Z\cdot \sigma}{E})^2$</p>
<p>From the formula you can see that to reduce the error, a larger sample size is required making for a more reliable test.</p>
| 549 |
statistics | Unbiased Estimators Poisson | https://math.stackexchange.com/questions/1502767/unbiased-estimators-poisson | <p>Suppose X ∼ Poisson(λ), instead of estimating λ, we
are interesting in estimating P(X = 0)^2 = e^−2λ. Show that δ(X) = (−1)X is an unbiased estimator for e^−2λ
. Is this a good estimator? Why?</p>
<p>I'm having a bit of trouble starting this problem. I know that equation for Bias and all, but how do I use it to cal... | <p>We have $\Pr(X=k)=e^{-\lambda}\frac{\lambda^k}{k!}$.
It follows that
$$E((-1)^X)=\sum_{k=0}^\infty (-1)^k e^{-\lambda}\frac{\lambda^k}{k!}.$$
Bring an $e^{-\lambda}$ out. We get
$$E((-1)^X)=e^{-\lambda}\sum_{k=0}^\infty (-1)^k\frac{\lambda^k}{k!}.$$
We recognize the sum after the $e^{-\lambda}$ as the usual power se... | 550 |
statistics | One tailed or two tailed | https://math.stackexchange.com/questions/1272083/one-tailed-or-two-tailed | <p>Ok so this the question:</p>
<p>An administrator at a medium-sized hospital tells the board of directors that, among patients received at the Emergency room and eventually admitted to a ward, the average length of time between arriving at Emergency and being admitted to the ward is 4 hours and 15 minutes. One of th... | <p>There is only one scenario. The administrator's statement has to be taken at face value and is the null hypothesis.</p>
<p>$H_0: t=4.25$ hours</p>
<p>The board member has an alternative hypothesis.</p>
<p>$H_1: t>4.25$ hours</p>
<p>This is one-tailed.</p>
| 551 |
statistics | Compare 2 algorithms by statistics | https://math.stackexchange.com/questions/1589763/compare-2-algorithms-by-statistics | <p>Let's suppose we have two different processes, each generating some amount of money $M$ every second.</p>
<p>$$0 \leq M \leq 1000$$</p>
<p>We run each process for $50\%$ of available time.</p>
<p>The question is how to compare the productivity (in money; per second) of these two processes if there is no informat... | <p>You have two samples - one from each method - with equal sample sizes (and big enough I assume) and you want to see which method generates <em>statistically</em> better results. This is a standard methodology with a confidence interval for the <em>difference of means</em> or a hypothesis testing again for the <em>di... | 552 |
statistics | Please Explain a simple Formula, calculating time in-between a call queue | https://math.stackexchange.com/questions/1521759/please-explain-a-simple-formula-calculating-time-in-between-a-call-queue | <p>I have a simple algebra formula, proven to work. But I need help in understanding why it works.</p>
<p>The Scenario: I work at a call center, and am trying to calculate the time free in-between calls. I have the 3 variables, provided by Live data:</p>
<ul>
<li>Staff Available (not on calls)</li>
<li>Staff Busy (on... | <p>Let's suppose any particular staff's workload can be represented by the busy staff and the available staff. Then, the staff spends $\frac{\text{Busy}}{\text{Total}}$ of the time taking calls and $\frac{\text{Available}}{\text{Total}}$ free. Since the calls are on average $5$ minutes long, if he take $5$ minutes taki... | 553 |
statistics | Joint probability distributions with continuous random variables | https://math.stackexchange.com/questions/1571251/joint-probability-distributions-with-continuous-random-variables | <p>Let $X, Y$ have the joint pdf $f(x, y)= 2, \quad 0 < y < x < 1$</p>
<p>I'm trying to calculate the marginal probability density functions, but I don't know which intervals I'm supposed to use. The source I'm learning from will interchange the following</p>
<p>$f_X(x) = \int_0^x$</p>
<p>$f_Y(y) = \int_y^1... | <p>In general, to get the marginal $f_X(x)$ you integrate $\int f(x,y)\mathop{dy}$ over all of $\mathbb{R}$. However, you have to consider the region where your joint density is nonzero.</p>
<p>For your particular joint density, $f(x,y)$ is zero when $y\ge x$ or $y \le 0$, so this would reduce to only integrating over... | 554 |
statistics | What is the probability that the minimum of _X1_, _X2_ and _X3_ is larger than 1? | https://math.stackexchange.com/questions/1559403/what-is-the-probability-that-the-minimum-of-x1-x2-and-x3-is-larger-than-1 | <p>Hello I found a problem on statistics which make me a bit confused. Could anyone help? Thanks!</p>
<p>Let <em>X1</em>, <em>X2</em>, <em>X3</em> be independent random variables following exponential
distribution with parameter <em>θ</em>=$5$. What is the probability that the minimum of <em>X1</em>,
<em>X2</em> and <... | <p>Well, one way is, if $M$ is the minimum of the three, then
\begin{align*}
P(M>1) &= P(X_1>1,X_2>1,X_3>1)\\
&= P(X_1>1)P(X_2>1)P(X_3>1)\tag{1}\\
&= e^{-5}e^{-5}e^{-5}\\
&=3.059023e-07,
\end{align*}
where (1) is true because of independence.</p>
| 555 |
statistics | Binomial and Geometric Variables- Finding expected values and variance | https://math.stackexchange.com/questions/1560322/binomial-and-geometric-variables-finding-expected-values-and-variance | <p>Suppose $X$ is a binomial random variable with parameters $(100, 1/3)$ and $Y$ is a geometric random variable with parameter 1/4.</p>
<p>(a) Find $E[(50 + X)^2]$.</p>
<p>(b) Find $Var(10 − 2Y )$.</p>
<p>a) I know that for a binomial random variable $E(X)=np$ and in this case $(n,p)=(100,1/3)$ so $E(X)=100/3$ from... | <p>a) You may already know that the variance of $X$ is $np(1-p)$, that is, $(100)(1/3)(2/3)$. </p>
<p>To find the expectation of $(50+X)^2$, expand the square, and use the linearity of expectation. We get $E(2500)+100E(X)+E(X^2)$.</p>
<p>To find $E(X^2)$, use the fact that $\text{Var}(X)=E(X^2)-(E(X))^2$.</p>
<p>b) ... | 556 |
statistics | How can I find $\operatorname{Cov}(X,Y)$ | https://math.stackexchange.com/questions/1562631/how-can-i-find-operatornamecovx-y | <p>$n=240$ trials with a $6$ sided dice</p>
<p>$X = \#5$'s</p>
<p>$Y = \#6$'s</p>
<p>How do I go about showing that $\operatorname{Cov}(X,Y) = -20/3$? I think I need to find $V(X+Y)$ but I'm not sure how. $V(X)=V(Y)=240* 1/6 * 5/6$. </p>
| <p>That is the covariance if the die rolled 240 times is fair (or the 240 dice rolled are fair). What if the die/dice is/are not fair?</p>
<p>X and Y are <a href="https://en.wikipedia.org/wiki/Binomial_distribution#Probability_mass_function" rel="nofollow">binomially distributed</a>:</p>
<p>$$P(X=x) = \binom{n}{x}(p_... | 557 |
statistics | Unbiased estimator of $\int_0^t \mu (s) ds$ | https://math.stackexchange.com/questions/1575235/unbiased-estimator-of-int-0t-mu-s-ds | <p>Let $\mu,\alpha_n:\mathbb R^+\to \mathbb R$ continuous function with $\mu$ bounded function.
Let $N^{(n)}$ the trajectory of a Poisson process with intensity $(\alpha_n \mu)(t)$.
Let $0=T_0^{(n)}<T_1^{(n)}<..$ jumps of $N^{(n)}$.</p>
<p>Let $M_n(t)=\sum_{i=1}^{N_t^{(n)}} \frac {1} {\alpha_n (T_i^{(n)})}$</p>
... | <p>You're close, it seems. The issue is that you've written
$$
E[E[M_n(t) | N^n_t = u] ] = E[M_n(t)]
$$
which is not true. What is true is that
$$
E[E[M_n(t) | N^n_t] ] = E[M_n(t)]
$$
With that in mind, you found that
$$
E[M_n(t)|N^n_t = u] = \frac{u}{\int_0^t \alpha \mu (s) \, ds} M(t)
$$
which implies that
$$
E[M_n(... | 558 |
statistics | Get precision of any decimal number | https://math.stackexchange.com/questions/1598451/get-precision-of-any-decimal-number | <p><em>(Sorry for the inconvenience related to the tags, please feel free to correct my post if it needs a better scope by adding some other tags).</em></p>
<p><strong>CONTEXT</strong></p>
<p>I have several (decimal) numbers shaped like this :</p>
<ul>
<li>1.081</li>
<li>289.089167</li>
<li>2.98</li>
<li>...</li>
</... | <p>Given that your input number is "$\color\red{n}.\color\green{m}$", the formula is "$\color\red{n}.\color\green{m}$"/"$\color\red{n}\color\green{m}$".</p>
<p>So in any scripting language, simply divide the rational number represented in the original string, by the integer number represented in the original string wi... | 559 |
statistics | How can I calculate the variance if the only thing I know is the difference between the random variables? | https://math.stackexchange.com/questions/1457954/how-can-i-calculate-the-variance-if-the-only-thing-i-know-is-the-difference-betw | <p>If I know $\alpha=x_2-x_1$ and $\beta=(x_3-x_1)+(x_3-x_2)$, how can I calculate the variance of $\{x_1,x_2, x_3\}$?</p>
| <p>$x_3-x_2=(x_3-x_1)-(x_2-x_1)$ allows you to easily obtain $x_k-x_1$. The shift by $x_1$ doesn't change the variance.</p>
| 560 |
statistics | Normal Distribution- Finding Expectations | https://math.stackexchange.com/questions/1564120/normal-distribution-finding-expectations | <p>Let X have a normal distribution with mean μ and variance $\sigma^2$. Find $E[X^3]$ (in terms
of μ and $\sigma^2$).</p>
<p>Im pretty sure that $μ= E[X]$ so to find $E[X^3]$ would i just split it up into $E[X*X^2]$ since i know $E[X]$ and $E[X^2]$ can be found from the variance formula?</p>
| <p><strong>Hint</strong>:</p>
<p>If $U$ has <em>standard</em> normal distribution then $\mathbb EU^3=0$. </p>
<p>This can be proved on base of symmetry (also $-U$ will have standard normal distribution so that $\mathbb EU^3=\mathbb E(-U)^3=-\mathbb EU^3$, and this implies $\mathbb EU^3=0$).</p>
<p>Observe that: $$U:... | 561 |
statistics | Finding joint distribution for the following | https://math.stackexchange.com/questions/1646896/finding-joint-distribution-for-the-following | <p>I am trying to do the following problem
Suppose that $X_1,...,X_n\stackrel{iid}\sim N(0,1)$. Define $$\bar{X}_k=\frac{1}{k-1}\sum_{i=1}^{k-1}X_i,\,\,\,\,\,\,\text{for }k=2,3,.....,n
$$</p>
<p>(i) What is the joint distribution of $(X_2-\bar{X}_2,X_3-\bar{X}_3,...,X_n-\bar{X}_n)$?</p>
<p>(ii) What is the distribut... | 562 | |
statistics | Mann–Whitney $U$test in $n$ dimensions | https://math.stackexchange.com/questions/1577278/mann-whitney-utest-in-n-dimensions | <p>This <a href="https://math.stackexchange.com/q/1576971/290307">https://math.stackexchange.com/q/1576971/290307</a> question reminded me of a yet unanswered question I had as a student.</p>
<p><a href="https://en.wikipedia.org/wiki/Mann%E2%80%93Whitney_U_test" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/... | <p>The problem of testing for the equality of $g$ group means is a well-discussed in nonparametric inference. Here are brief mentions of three methods in common use.</p>
<p>Solution (1) would be to look into the theory behind the Kruskal-Wallis test, which deals with exactly this issue. (K-W for two groups is precisel... | 563 |
statistics | Suggest an unbiased estimator for θ and provide an estimate for the standard error of your estimator. | https://math.stackexchange.com/questions/1647133/suggest-an-unbiased-estimator-for-%ce%b8-and-provide-an-estimate-for-the-standard-err | <p>If $Y_1, Y_2, \ldots , Y_n$ denote a random sample from an exponential distribution with mean $θ$, then $E(Y_i)=θ$ and $V(Y_i)=θ^2$. Thus, $E(\bar Y)=θ$ and $V(\bar Y)=θ^2/n$, or $σ_Y=θ/\sqrt{n}$. Suggest an unbiased estimator for θ and provide an estimate for the standard error of your estimator.</p>
<p>With these... | 564 | |
statistics | Finding a function that is a pivotal quantity of the MLE | https://math.stackexchange.com/questions/4989767/finding-a-function-that-is-a-pivotal-quantity-of-the-mle | <p>I'm confused on how to find a function <span class="math-container">$g_{(1)}$</span> of the MLE that is a pivotal quantity. I've never seen such notation for a function before. Here is the problem statement:</p>
<p>Let <span class="math-container">$Y_1$</span>,<span class="math-container">$Y_2$</span>,...,<span clas... | <p>It is true that <span class="math-container">$T=\frac{Y_{(1)}}{\theta}$</span> is a pivotal quantity. We must show that the distribution of <span class="math-container">$T$</span> does not depend on <span class="math-container">$\theta$</span>. To that end note that
<span class="math-container">$$
P(Y_{(1)}>y) = ... | 565 |
statistics | Scaling a uniform distribution - Probability | https://math.stackexchange.com/questions/314244/scaling-a-uniform-distribution-probability | <p>I just have a simple question on scaling a uniform distribution.
I know that uniform distribution has probability density of $1/(b-a)$ defined on the interval a to b. </p>
<p>My textbook says that we can scale the distribution to be between (0,1) and have a constant density of 1 by doing the following: </p>
<p>Su... | <p>We have $X$ which is a random variable of uniform distribution on $[a,b]$. Its expected value is the midpoint of the interval, $\frac{a+b}2$ (you can also verify it by the integral $\int_a^bx\cdot\frac1{b-a}dx$).</p>
<p>Now, at any possible experiment $X$ gets a concrete value, and this value is in $[a,b]$ with $1$... | 566 |
statistics | Law of Total Variance Using Three Different Variables | https://math.stackexchange.com/questions/4185243/law-of-total-variance-using-three-different-variables | <p>I have 3 random variables; X,Y, and Z. I am pretty familiar with law of total variance for two variables;</p>
<p><span class="math-container">$Var(X) = E[Var(X|Y)] + Var[E(X|Y)]$</span>.</p>
<p>(I'm sorry if the words are messy).</p>
<p>Recently, I read some papers, which state law of total variance using three vari... | <p>You have :
<span class="math-container">$$E\big(E[X|Y,Z]\big|Y,Z\big) = E(X)$$</span>
This is just the usual property, applied with the random variable <span class="math-container">$(Y,Z)$</span>, or if you want, you can rederive it like this :
<span class="math-container">\begin{align}
E[E[X|Y,Z]] &= E[ E [E [X... | 567 |
statistics | A question about $\sum_{i=1}^n x_i x_i^T = X^TX$ | https://math.stackexchange.com/questions/210920/a-question-about-sum-i-1n-x-i-x-it-xtx | <p>For a set of column vectors $x_1,\dots,x_n$, the identity shows that $\sum_{i=1}^n x_i x_i^T = X^TX$. I can show this by seeing the $(p,q)$ entry of the resulting matrix is $\sum_{i=1}^n (X^T)_{pi}X_{iq} = \sum_{i=1}^n x_{ip} x_{iq}$. Is there a quicker way of seeing this? and, does $xx^T$ have a special name?</p>
| <p>What you have written is true in general. If $A = \begin{pmatrix} a_1 & a_2 & \cdots &a_n\end{pmatrix}$ and $B = \begin{pmatrix} b_1 & b_2 & \cdots & b_n \end{pmatrix}$, then $$AB^T = \sum_{k=1}^{n} a_k b_k^T$$</p>
| 568 |
statistics | type II errors in Bartlett's test | https://math.stackexchange.com/questions/1136378/type-ii-errors-in-bartletts-test | <p>Various statistical techniques are based on the assumption that two samples have the same variance and Bartlett's test is meant to check for that. Using R I made a few tests with randomly generated numbers. For example:</p>
<p>bartlett.test(list(rnorm(10,sd=1),rnorm(10,sd=2)))</p>
<p>will perform a Bartlett test o... | 569 | |
statistics | Why is Sample Standard Deviation Biased? | https://math.stackexchange.com/questions/3145907/why-is-sample-standard-deviation-biased | <p>I've recently been going through Khan Academy Statistics, and I recently came across the fact that sample standard deviation is biased. Now, I know that there are many proofs online including several on math stack exchange, but I was wondering if someone could give me some intuition for why this happens instead of u... | <p>Reproduced from my argument in an <a href="https://artofproblemsolving.com/community/c7h216088_n__1_in_standard_deviation_formula" rel="nofollow noreferrer">AoPS thread</a>, also featuring a derivation of the sample variance:</p>
<p>The square root of this estimate for the variance is not an unbiased estimator of t... | 570 |
statistics | Sample, Random Variable, or Both? | https://math.stackexchange.com/questions/5038130/sample-random-variable-or-both | <p>I have seen <a href="https://math.stackexchange.com/questions/3622994/random-sample-random-variables-or-realizations-of-the-same-random-variable">this</a> question and it does not quite answer my question.</p>
<p>I have always had some confusion about the concept of a random variable, and how in statistics everythin... | <p>There is a tendency in probability to not talk about the sample space -- instead, one defines random variables and their distributions, and it is implied that the sample space will be "big enough" for them to work out.</p>
<p>The statement "Let <span class="math-container">$X_1, ... X_n$</span> be i.i... | 571 |
statistics | Calculating mode in a grouped frequency distribution | https://math.stackexchange.com/questions/1311506/calculating-mode-in-a-grouped-frequency-distribution | <p>How can I calculate the mode in a grouped frequency distribution when the largest frequency occurs in two or more classes?</p>
| <p>If the you have a frequency distribution where the largest frequency occurs in two or more classes, then we call it a <a href="https://en.wikipedia.org/wiki/Multimodal_distribution" rel="nofollow">multimodal distribution</a>. For example, if you set of data is $\{1,1,1,2,2,3,4,7,7,7,8\}$, the mode would be both of t... | 572 |
statistics | Variance of an unbiased estimator | https://math.stackexchange.com/questions/1902968/variance-of-an-unbiased-estimator | <p><span class="math-container">$T_1$</span> and <span class="math-container">$T_2$</span> are unbiased estimators of <span class="math-container">$\theta$</span>.
Then <span class="math-container">$T_3=aT_1+(1-a)T_2$</span> is also an unbiased estimator of <span class="math-container">$\theta$</span>.</p>
<p>If <span ... | <p>Denoting the variance as $\sigma^2$ and the expectation with an overline,</p>
<p>$$\sigma^2_{T_3}=\overline{(a(T_1-T)+(1-a)(T_2-T))^2}\\
=\overline{a^2(T_1-T)^2+2a(1-a)(T_1-T)(T_2-T)+(1-a)^2(T_2-T)}\\
=a^2\overline{(T_1-T)^2}+(1-a)^2\overline{(T_2-T)^2}\\
=a^2\sigma^2_{T_1}+(1-a)^2\sigma^2_{T_2}.$$</p>
<p>This is ... | 573 |
statistics | Degree of freedom and corrected standard deviation | https://math.stackexchange.com/questions/487954/degree-of-freedom-and-corrected-standard-deviation | <p>It is often said that degree of freedom causes the need for standard deviation formula to be corrected. When explaining degree of freedom, it is often said that when one knows the mean of the formula, only $n-1$ data are actually needed, as the last data can be determined using mean and $n-1$ data. However, I see th... | <p>Note that if the sample points $X_i$'s are iid normal then $\frac{1}{\sigma^2}\sum_{i=1}^n(X_i-\bar{X})^2$ follows $\chi^2_{n-1}$ distribution where $\bar{X}$ is the sample mean while $\frac{1}{\sigma^2}\sum_{i=1}^n(X_i-\mu)^2$ follows $\chi^2_n$ distribution, where $\mu$ is the population mean. The suffix of $\chi^... | 574 |
statistics | Pooled Variance Estimator efficiency. | https://math.stackexchange.com/questions/803199/pooled-variance-estimator-efficiency | <p>Apologies for the format of this question - I am new to this website. I am having trouble with part (c) of the question below, if anyone could assist that would be great.</p>
<p>Thanks in advance</p>
<p>Suppose that two independent random samples of size $n_1$ and $n_2$ observations are selected from normal popula... | <p>We have $$\begin{align} \text{Var}(S_{p1}^2) &=\text{Var}\left(\frac{(n_1-1)S_1^2 +(n_2-1)S_2^2}{n_1+n_2-2}\right) \\&= \left(\frac{n_1-1}{n_1+n_2-2}\right)^2\text{Var}(S_1^2)+\left(\frac{n_2-1}{n_1+n_2-2}\right)^2\text{Var}(S_2^2) \,\, \end{align}$$
and
$$\begin{align} \text{Var}(S_{p2}^2) &=\text{Var}\... | 575 |
statistics | Statistical comparisons for large sample sizes (n > 1000) | https://math.stackexchange.com/questions/3732609/statistical-comparisons-for-large-sample-sizes-n-1000 | <p>I am comparing the drug exposures across two different groups, consisting of 1000 simulated drug exposures per group. Drug exposures are continuous variables following a normal distribution.</p>
<p>I want to know if different doses yield a statistically significant difference in mean drug exposure across the two gro... | <h1>Effect of Sample Size on Power of One-Way ANOVA</h1>
<p>If you reduce sample sizes to 50 for each of the three treatment
groups, you may not have sufficient power to distinguish among
groups even if there are real differences among population means
of the three groups.</p>
<p>Just looking at the differences in samp... | 576 |
statistics | statistics, how to solve this problem | https://math.stackexchange.com/questions/4446346/statistics-how-to-solve-this-problem | <p>Two players, who have equal chances of winning each round, compete for a money prize. The first player to
win three rounds collects the total amount.
(a) If the game is interrupted when the score was 2 to 1, how should the players divide the stakes fairly?
(b) What if the game was interrupted when the score was 1 to... | 577 | |
statistics | Ask a question of Monte Carlo test | https://math.stackexchange.com/questions/4448901/ask-a-question-of-monte-carlo-test | <p>I am reading a book, it said:</p>
<blockquote>
<p>The expected power of a Monte Carlo test can be quite good even for
relatively small values of m. In simple situations (testing means, for
example), m = 99 may be a good choice. This allows the p-value to be
expressed simply in two decimal places. In more complicated... | 578 | |
statistics | Find $c$ for $P(\frac{5}{12}-c \le Y \le \frac{5}{12}+c)=\frac{1}{2}$ | https://math.stackexchange.com/questions/4449026/find-c-for-p-frac512-c-le-y-le-frac512c-frac12 | <p>Find <span class="math-container">$c$</span> for <span class="math-container">$P(\frac{5}{12}-c \le Y \le \frac{5}{12}+c)=\frac{1}{2}$</span></p>
<p>We also have that
<span class="math-container">$$f_Y(y) = k\sum_{i=0}^{\infty}y^i, y\in(1/3, 1/2) \\ \implies k\int_{1/3}^{1/2}\frac{1}{1-y}dy \implies k=\frac{1}{\log(... | <p>If the distribution is not "normal" then why would you think of applying tricks from normal distribution? . Secondly it is Chebycheff's <strong>Inequality</strong>. So applying it won't yield you any "equality" for <span class="math-container">$c$</span>.</p>
<p><span class="math-container">$$P(\... | 579 |
statistics | whats wrong with this approaches? | https://math.stackexchange.com/questions/1930501/whats-wrong-with-this-approaches | <p>1) In a population of men, the probability that a man’s left eye is of brown colour is p, and the probability that a man’s right eye colour is brown is also p. Therefore the probability that a man has at least one eye of brown colour is:
Pr(left eye brown or right eye brown)
= Pr(left eye brown) + P(right eye bro... | <p><strong>First Approach:</strong> You are counting the case where someone has two brown eyes twice. I think this principle is called Inclusion-Exclusion. You should have $2p-p^2$.</p>
<p><strong>Second Approach:</strong> The height and weight of someone aren't mutually exclusive. If they were, your approach would be... | 580 |
statistics | Usefulness of Variance | https://math.stackexchange.com/questions/31126/usefulness-of-variance | <p>I've had a look for intuitive explanations of the variance of an RV (e.g. <a href="https://math.stackexchange.com/questions/5392/intuitive-explanation-of-variance-and-moment-in-probability">Intuitive explanation of variance and moment in Probability</a>) but unfortunately for me, I still don't feel comfortable with ... | <p>The variance is easier to deal with in intermediate computations, because it doesn't have a square root. For example, if $X$ and $Y$ are independent, then $Var(X+Y) = Var(X) + Var(Y)$, which is a simpler formula than $SD(X+Y) = \sqrt{SD(X)^2 + SD(Y)^2}$. Basically, if you want to work in terms of standard deviation ... | 581 |
statistics | Relation between repeat number in coin toss | https://math.stackexchange.com/questions/128892/relation-between-repeat-number-in-coin-toss | <p>I am trying to establish correlation between tossing of coins and occurring of repeats.</p>
<blockquote>
<p>Coin is flipped 10 time as follows:</p>
<p><span class="math-container">$${\rm H.T.H.H.H.T.H.T.T.T. }$$</span></p>
<p>After each repeat occurring I have put (R) as follows:</p>
<p><span class="math-container">... | <p>We assume the coin is fair. Then after the first toss, the probability of a repeat is $1/2$. For $k>1$, the event there is an R at position $k$ is independent of previous locations of the R's. So if we toss the coin $n$ times, the number of R's has <em>binomial</em> distribution, where the number of trials is $n-... | 582 |
statistics | Question on $p$-value for two-sided test | https://math.stackexchange.com/questions/139553/question-on-p-value-for-two-sided-test | <p>I'm doing some revision here and I think one of the answers in my notes is wrong. It says on my notes the answer is D). Here's the question:</p>
<blockquote>
<p>A researcher conducted a large sample two-sided test of the null hypothesis that <span class="math-container">$u = 100$</span>. She reports a <span class="m... | <p>The factor of $2$ doesn't enter into comparing the $p$-value and the significance level. A $p$-value of $0.034$ means that if the null hypothesis were true data at least as extreme as the observed data would have been observed with probability $0.034$. That's enough to reject the null hypothesis at a significance le... | 583 |
statistics | Combining errors given standard deviation | https://math.stackexchange.com/questions/285101/combining-errors-given-standard-deviation | <blockquote>
<p>I find the mass of liquid in a container by using</p>
<blockquote>
<p>mass of liquid = mass of (container+liquid) - mass of container</p>
</blockquote>
<p>My measurements are subject to an error with mean zero and standard deviation 0.7g. Find the standard deviation of the error in the calculated mass o... | <p>Variance(mass of liquid) = Variance(mass of (container+liquid)) + Variance(mass of container) if the measurements of mass of (container+liquid) and mass of container are independent. </p>
| 584 |
statistics | Expected Value Calculation | https://math.stackexchange.com/questions/232669/expected-value-calculation | <blockquote>
<p>Let X and Y be discrete random variable with joint pdf <span class="math-container">$f(x,y) = 4/5xy$</span> if <span class="math-container">$x = 1,2$</span> and <span class="math-container">$y = 2,3$</span> and zero otherwise. Find:</p>
<p>E(Y)</p>
</blockquote>
<p>Basically I found the marginal pdf and... | <p>It's generally a good idea to avoid notation like $4/5xy$ because it's ambiguous with respect to the order of operations; my tendency would have been to interpret it as $(4/5)xy$, whereas you apparently intended $4/(5xy)$.</p>
<p>The notation $y=2,3$ is also suboptimal, both because $y$ is equated to two different ... | 585 |
statistics | How to obtain this particular probability distribution and standard deviation? | https://math.stackexchange.com/questions/1628596/how-to-obtain-this-particular-probability-distribution-and-standard-deviation | <p>Currently, I am studying statistics as an undergraduate. Our lecture today finished with information about obtaining probability distributions and expected values (using binomial and geometric distributions).</p>
<p>Later that day, after throwing some snow with some friends, I figured it would be neat to use what I ... | 586 | |
statistics | uniform distribution for probability | https://math.stackexchange.com/questions/1712484/uniform-distribution-for-probability | <blockquote>
<p>The city needs to perform some road maintenance and will rent excavator machines from a company. Each excavator will work for at least one hour and no more than <span class="math-container">$4$</span> hours in a day. The working time is evenly distributed.</p>
<p>Given that a excavator has already w... | <p>As we have
P (A|B)=P [A,B]/P (B)
I,e
P [x>3.5|x>2.5]=P [x>3.5,x>2.5]/P [x>2.5]
=P [x>3.5]/P [x>2.5]
since intersection of [x>3.5,x>2.5]=[x>3.5]</p>
| 587 |
statistics | Which subject are these students best? (basic high school statistical question) | https://math.stackexchange.com/questions/2332766/which-subject-are-these-students-best-basic-high-school-statistical-question | <p>It's a very basic high school statistical question, but I'm struggling to solve it.</p>
<p>Suppose I have a school with <span class="math-container">$287$</span> students and each one made a test with <span class="math-container">$50$</span> questions (multiple choice questions with <span class="math-container">$5$<... | <p>You define failure as scoring a zero on the test. Then of course, the students are more likely to score a zero if there are fewer questions. Let us say that a test has $n$ questions, and $5$ options per question. Then the probability of getting all the answers wrong is $(\frac{4}{5})^n$. So the probability of "not f... | 588 |
statistics | How to evaluate which test method is better? | https://math.stackexchange.com/questions/3728841/how-to-evaluate-which-test-method-is-better | <p>The short form of my question is :</p>
<p>Test A shows a useful result after one trial at 16% chance, two trials at 41,7% chance and after three trials with 75% chance. Is it better or worse to use it compared to test B which needs at least two trials and shows a useful result after two trials at 33,3% chance and af... | 589 | |
statistics | Amount of trials until all marbles have shown in a box of different marbles. | https://math.stackexchange.com/questions/3739900/amount-of-trials-until-all-marbles-have-shown-in-a-box-of-different-marbles | <p>We have 81.000 different marbles in a box. With every grab, I get 30 different marbles out of the box. After this, the 30 marbles have to be thrown back into the box so that there are again 81.000 different marbles.</p>
<p>How often do I have to grab into the box until I have statistically seen at least one time eve... | <p>The question is a bit vague, but one interpretation is that we would like to know how many grabs are required, on average, to have drawn each marble at least once.</p>
<p>A simplifying assumption seems appropriate here. I think we may as sell assume that grabbing a batch of <span class="math-container">$30$</span> ... | 590 |
statistics | How to choose size of samples to determine if set contains elements | https://math.stackexchange.com/questions/4725391/how-to-choose-size-of-samples-to-determine-if-set-contains-elements | <p>I have sets of elements (lets call each <code>S</code>). The elements in each <code>S</code> can only ever be <code>black</code> or <code>white</code>. I also have a test that determines if an element is <code>black</code> or <code>white</code>, with some false-positive/false-negative rate <code>FP</code>/<code>FN</... | <p>The problem of determining the sample size to estimate a proportion of the population is a typical textbook example. For example, if we think of each set <span class="math-container">$S$</span> as a coin, the task is to determine how many times should the coin be tossed to check whether it is unbiased. Or as another... | 591 |
statistics | How to calculate E(x^2) in order to find variance | https://math.stackexchange.com/questions/3202867/how-to-calculate-ex2-in-order-to-find-variance | <p>Lets say that we are trying to find the variance of a coin with a <span class="math-container">$0.6$</span> probability of heads flipped <span class="math-container">$n$</span> times.
(Binomial with <span class="math-container">$p=0.6$</span>)</p>
<p>Given that the equation to find variance is <span class="math-con... | <p>Note the flips are independent, so <span class="math-container">$1$</span>-if-heads-<span class="math-container">$0$</span>-if-tails are <span class="math-container">$n$</span> independent variables across the flips. If you calculate the variance for one flip, the variance for <span class="math-container">$n$</span>... | 592 |
statistics | Precise mathematical translation of the 68–95–99.7 rule?(Not a proof!) | https://math.stackexchange.com/questions/493046/precise-mathematical-translation-of-the-68-95-99-7-rulenot-a-proof | <p>The rule:</p>
<p>In statistics, the 68–95–99.7 rule, also known as the three-sigma rule or empirical rule, states that nearly all values lie within 3 standard deviations of the mean in a normal distribution.</p>
<p>About 68.27% of the values lie within 1 standard deviation of the mean. Similarly, about 95.45% of t... | <p>If $X$ is a normally distributed random variable, then
$$\Pr\left(\left|\frac{X-\mu}{\sigma}\right|\le 1\right)\approx 0.6826.$$
Here $\mu$ is the population mean, and $\sigma$ is the population standard deviation.</p>
<p>Similar facts hold for the other two numbers you mentioned. </p>
<p>If we do repeated <stron... | 593 |
statistics | Calculate variance of a subset | https://math.stackexchange.com/questions/2238086/calculate-variance-of-a-subset | <p>I have two sets of products A + B in the same product category. I have the overall and A's N, mean and variance. Can I calculate the variance of B with this?</p>
<p>I noticed <a href="https://stats.stackexchange.com/questions/29170/how-to-compute-standard-deviation-of-difference-between-two-data-sets">this</a> answ... | <p>Let us denote the sample variance of Product A items as $$s_A^2 = \frac{1}{N_A - 1} \sum_{i=1}^{N_A} (A_i - \mu_A)^2,$$ and the sample variance of product B items as $$s_B^2 = \frac{1}{N_B - 1} \sum_{i=1}^{N_B} (B_i - \mu_B)^2,$$ where $N_A$, $N_B$ are the sample sizes for each product; $A_1, A_2, \ldots, A_{N_A}$ a... | 594 |
statistics | Estimation of shape area given random points inside it | https://math.stackexchange.com/questions/4732599/estimation-of-shape-area-given-random-points-inside-it | <p>I have a shape <span class="math-container">$S$</span> of area <span class="math-container">$A$</span>, but I do not know what <span class="math-container">$A$</span> is. However, I do have the coordinates of <span class="math-container">$N$</span> points (the coords are <span class="math-container">$(x_i,y_i)$</spa... | <p>The area of any shape can be written: <span class="math-container">$A=\sigma^2\gamma(S)$</span> where <span class="math-container">$\gamma$</span> is a scale factor and <span class="math-container">$\sigma$</span> is the 2D radius of gyration. <span class="math-container">$$\hat A=s^2\gamma(S)$$</span></p>
<p>Where ... | 595 |
statistics | If mean=mode, mean=median? | https://math.stackexchange.com/questions/2227286/if-mean-mode-mean-median | <blockquote>
<p>If the mean of a set of data is equal to the mode, is it always equal to the median? Explain why or why not.</p>
</blockquote>
<p>I would guess no, but after generating countless sets I cannot find anything that contradicts this</p>
| 596 | |
statistics | Why is $P(X>C) = E[1\{X>C\}]$ | https://math.stackexchange.com/questions/2244758/why-is-pxc-e1-xc | <p>This was found in a line from the proof of the law of large numbers. the $1\{X>C\}$ is the usual indicator function. </p>
| <p>Define $Y = I\{X>c\}$ then
$$
EI\{X>c\} =EY = 1P(X>c)+0P(X\le c) = P(X>c).
$$
You can see that $Y \sim Bernoulli ( P(X>c))$.</p>
| 597 |
statistics | Salary scheme, is HR is cheating? | https://math.stackexchange.com/questions/2227472/salary-scheme-is-hr-is-cheating | <p>I like to ask you about salary payment scheme by a company. Here is interesting points of this company salary scheme. </p>
<p><strong>New company payroll scheme</strong></p>
<ol>
<li>Salary is credit on every 25th of every month. </li>
<li>Number of working days is 30 days flat. </li>
<li>A few people are joined o... | <p>Of course HR is cheating - that's their job.</p>
| 598 |
statistics | Fair game question | https://math.stackexchange.com/questions/2165837/fair-game-question | <p>an urn contains 25 balls 40% of which are green. a contestant reaches in the urn to choose three balls the contestant will win 200 if he or she selects a green ball but will lose 120 for any other colour. is this a fair game</p>
<p>A) the ball is replaced after each draw
B) the ball is not replaced after each draw<... | <p>Whether balls are replaced or not is irrelevant in terms of <em>expected</em> winnings, since the expectation of what you replace obviously coincides with the expectation of what you draw (note that replacement does increase the variance, however). So, whether you replace balls or not, the expected winning per ball ... | 599 |
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