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probability distributions | Expected distribution of random draws | https://stats.stackexchange.com/questions/3650/expected-distribution-of-random-draws | <p>I have a two part question;</p>
<p>First Part:</p>
<p>I have an urn with 20 balls, 2 of those balls are purple, and I pull out 6 balls at random. I witness 100 realizations of this process. </p>
<p>Given the observed frequency at which I drew purple balls, how do I determine if I am really pulling balls out at ra... | <p>The expected frequency of observing $k$ purple balls in $d$ draws (without replacement) from an urn of $p$ purple balls and $n-p$ other balls is obtained by counting and equals</p>
<p>$$\frac{{p \choose k} {n-p \choose d-k} }{{n \choose d}}.$$</p>
<p>Test a sample (of say $100$) such experiments with a chi-squared... | 500 |
probability distributions | Find cumulative probability from given formula | https://stats.stackexchange.com/questions/192926/find-cumulative-probability-from-given-formula | <p>Given a formula to calculate instantaneous probability of an event.<br/>
<code>f(i) = 0.0222 * e ^ (-i / 11.5)</code></p>
<p>For instance 0.0222 * e ^ (-4 / 11.5) is the probability of the event occurring exactly during the fourth months given that it hasn’t happened before. Calculate the cumulative probability ... | <p>The cumulative distribution is simply the integral of the pdf</p>
<p>$F(i)=\int_{-\infty}^i f(k) dk$</p>
<p>in this case</p>
<p>$F(i)=\int_{-\infty}^i \frac{1}{11.5}e^{-\frac{k}{11.5}} dk$</p>
<p>$F(i)=-\frac{11.5}{11.5}e^{-\frac{i}{11.5}} - -\frac{11.5}{11.5}e^{-\frac{-infty}{11.5}} $</p>
<p>$F(i)=1-e^{-\frac{... | 501 |
probability distributions | Assessing the validity of a PMF? | https://stats.stackexchange.com/questions/485938/assessing-the-validity-of-a-pmf | <p>How would one go about solving the following given that the function h(x) isn’t provided in the question? I’m at a loss on where to begin.</p>
<p>Suppose h(x) is such that h(x) > 0 for x = 1,2,3,...,I. Argue that <span class="math-container">$p(x) = h(x)/ \sum_{i=1}^I h(i)$</span> is a valid pmf</p>
| <p>The fact that <span class="math-container">$h(x)$</span> isn't provided is a strong hint: it <em>doesn't matter</em> what <span class="math-container">$h(x)$</span> is except that it's always positive (<span class="math-container">$\geq 0$</span> would also be ok).</p>
<p>First, why is <span class="math-container">$... | 502 |
probability distributions | Joint distributions and Function of a random variable | https://stats.stackexchange.com/questions/231094/joint-distributions-and-function-of-a-random-variable | <p>In a probability distribution, is it true that $XX$ is NOT $X^2$. That is, $XX$ is a joint distribution of $X$ and $X$ and $X^2$ is a function of $X$?</p>
| <p>I think it is a notation problem: what does $XX$ represent? Note $XX$ is not a widely used notation.</p>
<p>Are you trying to use $XX$ to represent $2$ outcomes from two random events? or Are you tying to use $XX$ to represent a product of two random variables.</p>
<p>If $XX$ represents $2$ outcomes, then the dist... | 503 |
probability distributions | How to find difference between multiple probability distributions? | https://stats.stackexchange.com/questions/232259/how-to-find-difference-between-multiple-probability-distributions | <p>I have few <code>vectors</code> (of length <code>1000</code>) representing <strong>frequency of 1000 elements for different situations</strong>.</p>
<p>e.g. Vector for situation 1 is </p>
<p><code>s1=(12, 0, 3, 4, 0, ...., 10)</code></p>
<p>and is of length <code>1000</code> (as there are <strong>1000 distinct el... | 504 | |
probability distributions | Is there a constructive approach to creating a distribution achieving the Chebyshev lower bound? | https://stats.stackexchange.com/questions/236248/is-there-a-constructive-approach-to-creating-a-distribution-achieving-the-chebys | <p>I came across the following question and am wondering if there's a simple way to cook up a distribution achieving the Chebyshev lower bound:</p>
<p>Suppose $X$ has $\mu_X = \sigma^2_X = 9$. Define the lower bound for</p>
<p>$$\mathbb{P}[3 \leq X \leq 15]$$</p>
<p>Chebyshev tells us the answer is $\frac34$.</p>
<... | 505 | |
probability distributions | Is there any way of estimating the value of a variable when you know its probability distribution? | https://stats.stackexchange.com/questions/246951/is-there-any-way-of-estimating-the-value-of-a-variable-when-you-know-its-probabi | <p>I have one question regarding the estimation of an unknown variable.</p>
<p>Is there any way of estimating the value of a variable when you know its probability distribution?
In this case I have a variable which is distributed on the interval (1, 2), with a 25% probability of 2, and uniformly distributed otherwise.... | <p>A random variable doesn't have a single unique value (unless it's <a href="https://en.wikipedia.org/wiki/Degenerate_distribution" rel="nofollow noreferrer">degenerate</a>, and your variable isn't). On the contrary, random variables exist to provide a mathematical formalism for something that might take on any of a v... | 506 |
probability distributions | Factorizing a probability distribution | https://stats.stackexchange.com/questions/247536/factorizing-a-probability-distribution | <p>I am trying to read a paper on factor graphs and coming from a cs background I am a little lost on the following proposition.</p>
<p>Not all distributions can be factored into a product of clique potentials.</p>
<p>The example they cite is the uniform distribution over binary vectors with an even number of ones. W... | 507 | |
probability distributions | Finding the number of trials required for a value to be within a certain range in a discrete distribution at a certain probability | https://stats.stackexchange.com/questions/250765/finding-the-number-of-trials-required-for-a-value-to-be-within-a-certain-range-i | <p>In a survey of random questions, Jane can reply with either of the two possible answers; 'Yes' or 'No'.
The theoretical probability of Jane selecting 'Yes' for any random question in the survey is (p). </p>
<p>0 < (p) < 1 and (p) is a decimal with two decimal places (i.e. (p) can be 0.00, 0.01, 0.02, 0.03, ..... | 508 | |
probability distributions | How to model cache hits | https://stats.stackexchange.com/questions/260692/how-to-model-cache-hits | <p>I'd like to model the behavior of a server A, which caches results from an upstream server B. Queries sent to server A will be forwarded to server B if the former has not already seen this query for an entry. </p>
<p>Relaxations and parameters:</p>
<ul>
<li>There are <code>x</code> total entries in server B, which... | 509 | |
probability distributions | Determine $x,x∈R^+$ such that $φ(x)=0,9505$ | https://stats.stackexchange.com/questions/262191/determine-x-x%e2%88%88r-such-that-%cf%86x-0-9505 | <p><em>I tried to use the definition:</em>
$$\displaystyle φ(x) = \frac{1}{\sqrt{2\pi}}
\int_{-\infty}^x e^{-{s^2}/{2}}\,\mathrm ds$$</p>
<p><em>So, according to <a href="http://es.symbolab.com/solver/step-by-step/%5Cint%20e%5E%7B-%5Cfrac%7Bx%5E2%7D%7B2%7D%7D" rel="nofollow noreferrer">this</a> site:</em>
$$\int \:e^{... | <p>Tables of $\Phi(x)$ can be found in many textbooks, on-line (e.g. <a href="https://www.mathsisfun.com/data/standard-normal-distribution-table.html" rel="nofollow noreferrer">here</a>), etc, and you simply look in the table for the value of $x$ for which $\Phi(x)$ equals $0.9505$. Alternatively, there are various on-... | 510 |
probability distributions | Tricky Question re Customer Purchase Probabilities | https://stats.stackexchange.com/questions/183714/tricky-question-re-customer-purchase-probabilities | <p>I would like to calculate some parameters relating to customer purchasing in a retail situation. </p>
<p>I have some basic information which I can use:</p>
<p>Customer visit frequency in the form of probability distribution (I can generate Excel poisson tables using average visit frequency and these work well) for... | <p>what you are looking for is the sum of independent random variables. This can be calculated either by convolution or by using moment generating functions.</p>
<p><a href="https://www.stat.wisc.edu/courses/st311-rich/convol.pdf" rel="nofollow">discrete convolution formula</a>
so in excel you should be able to do it... | 511 |
probability distributions | Probability of median > 1.5 | https://stats.stackexchange.com/questions/194123/probability-of-median-1-5 | <p>I was asked a probability question: </p>
<p>Given three numbers i.i.d as $\text{uniform}(0,2)$, what is the probability of the median greater than $1.5$?</p>
<p>My hunch is that each number has $P(X > 1.5) = (2-1.5)/(2-0) = 0.25$, the probability of $\text{median}> 1.5$ is equivalent to "at least two of the... | <p>I would suggest looking at the order statistics. If your three observations $X_1, X_2, X_3$ follow a $Uniform(0,2)$, then the median is $X_{(2)}$, where $X_{(2)}$ is the second order statistic. Your problem then becomes much simpler, as you want to find $P(X_{(2)}>1.5)$. You can easily look up the PDF of order st... | 512 |
probability distributions | Looking for a distribution with very specific properties | https://stats.stackexchange.com/questions/210039/looking-for-a-distribution-with-very-specific-properties | <p>I'm looking for a <strong>continuous</strong> distribution which I can parameterize such that </p>
<ol>
<li>The expected value is roughly zero</li>
<li>The expected maximum given $x$ draws from that distribution is only very weakly increasing in $x$</li>
<li>$Prob(\max $ of $ x $ draws $ > 0)$ is small for "lar... | <p>How about $y \tilde{} \mathscr{N}(-a, fa)$ for small $a>0$ and $f>1$?</p>
<ol>
<li>E$[y]$ = $-a$.</li>
<li>E[max of $x$ draws] = $-a + 2fa\sqrt{\ln x}$ = $a[2f\sqrt{\ln x}-1]$. </li>
<li>Probability of maximum > 0 is similarly increasing in x, and controlled by $f$. It is something like $f_Z(z) = x F_Y(y)^{x-... | 513 |
probability distributions | How to properly initialize a stochastic vector? | https://stats.stackexchange.com/questions/213644/how-to-properly-initialize-a-stochastic-vector | <p>I'd like create a stochastic vector $$\mathbf{v} = (v_1, \dots, v_n)$$ of length $n$, so that its elements are assigned weights according to a given parameter ("entropy"): the weights are either uniformly distributed with $p_i = 1/n$, or there's a "bump" at a particular position and other elements decay to 0 quickly... | 514 | |
probability distributions | How Many Random Choices Before They Have All Been Picked About The Same # Of Times? | https://stats.stackexchange.com/questions/221745/how-many-random-choices-before-they-have-all-been-picked-about-the-same-of-tim | <p>Given a set of numbers 1 through N, how many times do I have to randomly pick a number before all of the numbers have been picked a roughly equivalent (within 5% of each other) amount of times, and each one has at least been picked 100 times?</p>
<p>For example:
Given numbers 1-10, how many times do I have to rando... | <p>Multinomial problems can be tricky. But when the number of observations grows large, approximations can work very well. This post explains how to use mental arithmetic (or, at worst, the back of a napkin) to obtain a reasonable answer. The beauty of this approach lies in how one can solve challenging statistical ... | 515 |
probability distributions | How can I determine if a weighed random function is working as expected? | https://stats.stackexchange.com/questions/223217/how-can-i-determine-if-a-weighed-random-function-is-working-as-expected | <p>I am writing a series of program utilities which heavily utilize weighted random functions, where the user defines a probability density function as a piecewise curve with a series of <code>(x, y)</code> coordinates and gets back numbers which conform to that function. What I've been doing so far to verify the worki... | 516 | |
probability distributions | In a statitical experiment involving independent coin tosses, what is the number of heads required to get $m$ tails? | https://stats.stackexchange.com/questions/136794/in-a-statitical-experiment-involving-independent-coin-tosses-what-is-the-number | <p>I know that the solution is a negative binomial distribution. However, I was a looking for a proof for the same along the following lines. If the number of coin tosses is fixed to be $n$, then the distribution of number of heads follows a binomial distribution, that is
\begin{equation}
P(N_h = n_h) = \frac{n!}{n_h!(... | <p>Assume that we have observed $m$ tails. The last coin toss must have given us a tail - otherwise we would have stopped earlier. We can now count in how many ways this can occur with $n_h$ heads. Thus, we have $n_h + m$ tosses, and the last one is fixed (tail). We then have to place $m - 1$ tails in $n_h + m - 1$ pla... | 517 |
probability distributions | Given hit probability and number of projectiles, randomly determine number of hits | https://stats.stackexchange.com/questions/155412/given-hit-probability-and-number-of-projectiles-randomly-determine-number-of-hi | <p>I'm working on a game where I'd like damage dealt to a target to be randomized, but having troubles working out how to go about it.</p>
<p><code>N</code> projectiles are fired at a target with probability, <code>p</code>, that each missile will hit the target and damage dealt is based on the number of projectiles t... | <blockquote>
<p>N projectiles are fired at a target with probability, p, that each missile will hit the target and damage dealt is based on the number of projectiles that hit.</p>
<p>I'd like to randomly generate the number of projectiles that hit. Fractional values would likely be considered as partial hits for this p... | 518 |
probability distributions | How to work out covariance using the expected value of x,y and xy | https://stats.stackexchange.com/questions/156165/how-to-work-out-covariance-using-the-expected-value-of-x-y-and-xy | <blockquote>
<p>Cov(x,y) if e(x) = 2.60 and e(y) = 2.35 and e(xy) = 7.06</p>
</blockquote>
<p>i'm confused because all i know is
$$cov(x,y) = \text{correlation of xy}*sqrt(x)*sqrt(y) $$</p>
| 519 | |
probability distributions | How to determine if random variables are distributed according to a multivariate normal distribution? | https://stats.stackexchange.com/questions/173440/how-to-determine-if-random-variables-are-distributed-according-to-a-multivariate | <p>Suppose $(x_1, x_2, x_3)\sim N(\mu, \Sigma)$ where $\mu\in\mathbb{R}^3$ and $\Sigma$ is a $3\times 3$ covariance matrix, are the variables $A = x_1 + x_2$ and $B = x_2 + x_3$ necessarily distributed according to a multivariate normal distribution? That is, do there exist $\nu\in\mathbb{R}^2$ and a $2\times 2$ covari... | <p>Yes and here is how to see it. Write $A$ and $B$ as</p>
<p>$$\begin{bmatrix} A \\B \end{bmatrix}=\begin{bmatrix} 1 & 1 & 0 \\0 & 1 &1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$$</p>
<p>and exploit the fact that <a href="https://en.wikipedia.org/wiki/Multivariate_normal_distribut... | 520 |
probability distributions | Why doesn't 1/2 always equal the value of X's CDF at E[X]? | https://stats.stackexchange.com/questions/177178/why-doesnt-1-2-always-equal-the-value-of-xs-cdf-at-ex | <p>Doesn't exactly half the probability fall on either side of the mean?</p>
| <blockquote>
<p>Doesn't exactly half the probability fall on either side of the mean?</p>
</blockquote>
<p>No!</p>
<p>The value half the probability falls either side of (at least with continuous distributions) is called the <strong><em>median</em></strong> (with discrete distributions you have to rephrase as somet... | 521 |
probability distributions | pdf and their relation to moments | https://stats.stackexchange.com/questions/484434/pdf-and-their-relation-to-moments | <p>I read that gaussian distribution is only defined by 2 moments i.e., mean and variance and can be only defined by these moments. Do we have distributions defined by just mean? Even for that matter, are there distributions defined by first three moments or say infinite moments. Can someone throw examples for such?</p... | <p>To reframe the question a bit, let us recall that (raw or absolute) moments of a distribution <span class="math-container">$p$</span> are quantifies related to expectations (when defined): <span class="math-container">$E_p[(x-a)^\alpha]$</span> or <span class="math-container">$E_p[|x-a|^\alpha]$</span> and they can ... | 522 |
probability distributions | Probability distribution of random variables | https://stats.stackexchange.com/questions/271458/probability-distribution-of-random-variables | <p>The SAT is used as an aid in determining college admissions. This test is a multiple choice test. To discourage random guessing, points are subtracted for wrong answers. Each question has 5 possible answers, and the test taker must pick one answer or choose not to answer the question. One point is awarded for each c... | <p>So, If I understand correctly, you are asking for the probability distribution of the number of points for a <em>single question.</em></p>
<p>The use of a Bernoulli (0 or 1) random variable can help here. Let $Y \sim Bern(p)$, where $p$ is the probability of getting a question correct. Then the number of points ear... | 523 |
probability distributions | Central limit theorem for non-identical distributed random variables | https://stats.stackexchange.com/questions/461272/central-limit-theorem-for-non-identical-distributed-random-variables | <p>Suppose <span class="math-container">$x_i, i=1, ..., n$</span> are independently distributed with mean <span class="math-container">$0$</span> and variance <span class="math-container">$\sigma^2$</span>. Then <span class="math-container">$\frac{1}{\sqrt n}\sum_{i=1}^n a_ix_i$</span> converges to a normal distributio... | <p>Necessary and sufficient conditions for a CLT of variables that are independent but not iid are known: the Lindeberg-Feller CLT.</p>
<p>The conditions for a CLT of an independent zero-mean, sequence <span class="math-container">$y_n$</span> (or triangular array <span class="math-container">$y_{in}$</span> to converg... | 524 |
probability distributions | Phase space distribution of heads and tails (coin toss) | https://stats.stackexchange.com/questions/70670/phase-space-distribution-of-heads-and-tails-coin-toss | <p>So I have the equation </p>
<p>$$h(t) = 1 + vt - \frac{1}2 gt^2 \pm sin(\omega t) $$</p>
<p>to describe the motion of a flipped coin. It is just a kinematics equation with an angular component added to it, where $h$ is the height of the coin, $v$ is the upwards velocity of the coin, $\omega$ is the angular velocit... | 525 | |
probability distributions | What distribution would be expected for number rapists ( vs number of victims)? | https://stats.stackexchange.com/questions/71641/what-distribution-would-be-expected-for-number-rapists-vs-number-of-victims | <p>This is a horrible question to ask. But it would be useful to know (rather than someone spouting an <em>opinion</em> that 99.999% are/are not.).</p>
<p>It has been estimated that 18.3% of women will be raped at some point in their life time (<a href="http://www.cdc.gov/ViolencePrevention/pdf/sv-datasheet-a.pdf" rel... | <p>How about a zero-inflated model?</p>
<p>It seems reasonable to assume that the population contains a (hopefully large) number of good folks who never rape anyone, mixed together with a subpopulation of rapists that attack people according to some other as-of-yet-unspecified distribution. </p>
<p>Fortunately for me... | 526 |
probability distributions | Calculating the probability of an inequality with two random variables | https://stats.stackexchange.com/questions/78030/calculating-the-probability-of-an-inequality-with-two-random-variables | <p>I am analyzing a timing circuit I designed, and I need to calculate the probability of a certain event (bit error). For example, I have derived this equation:</p>
<p>$(1 + x) / d < 1 / M$,</p>
<p>where x is a random variable with a normal distribution, d is a random variable with uniform distribution, and M is ... | <p>I think that your ratio is slash distributed. </p>
<p><a href="http://en.wikipedia.org/wiki/Slash_distribution" rel="nofollow">http://en.wikipedia.org/wiki/Slash_distribution</a></p>
<p>It should be possible to derive the probability $<1/M$ from there. Another option perhaps is bootstrapping, where you would ta... | 527 |
probability distributions | Expected number of trials before k successes where multiple successes can occur at each trial | https://stats.stackexchange.com/questions/94891/expected-number-of-trials-before-k-successes-where-multiple-successes-can-occur | <p>Let's say there are $N$ balls, $l$ of them have unique colours, $N-l$ of them are black.</p>
<p>$D$ people uniformly randomly sample $n$ balls without replacement. Each person's sampling is independent of the others'. So they take $n$ balls, make copies of them, and then put the originals back for the next person.<... | 528 | |
probability distributions | Cumulative probability of 2 independent events | https://stats.stackexchange.com/questions/97539/cumulative-probability-of-2-independent-events | <p>I’m trying to find the cumulative probability of 2 independent events (at least I think its cumulative probability). </p>
<p>I will miss work on rainy days (event 1) where my car won’t start (event 2).
The probability that my car won’t start on any given day is 20%.
The probability of rain on any given day in a mo... | 529 | |
probability distributions | How do I determine/compute a cutoff for chance level/not chance level? | https://stats.stackexchange.com/questions/102643/how-do-i-determine-compute-a-cutoff-for-chance-level-not-chance-level | <p>I am in the process of describing my research design for my dissertation and ran into a roadblock. In my design, I am converting 20 y/n responses from 190 participants to two dichotomous groups: 1). Chance level and 2). Above chance level.</p>
<p>If chance level is 50% or 10 responses correct/incorrect, how do I de... | <p>Here's one possibility:</p>
<p>If you assume the answers are like independent coin-flips, you can work out the probability of getting any number correct. For example if the answers were 50-50 coin flips the change of getting at least 15 correct is about 2%. You might say "well, that's pretty unlikely, let's say th... | 530 |
probability distributions | Proof of conditional probability | https://stats.stackexchange.com/questions/104351/proof-of-conditional-probability | <p>$X$ and $Y$ are real-valued random variables such that the distribution of $(X,Y)$ is absolutely continuous with density function $p$ and let $p_x$ denote the marginal density function of $X$. Suppose that there exists a point $x_0$ such that $p_x(x_0) > 0$, $p_x$ is continuous at $x_0$, and for almost all $y$, $... | <p>Recalling the definition of conditional density we have
$$
\Pr(Y\in A\mid X=x_0) = \int_A f_{Y\mid X}(y\mid x_0)\,dy = \frac{1}{f_X(x_0)} \int_A f_{X,Y}(x_0,y)\,dy \, .
$$
For $\epsilon>0$, consider
$$
\Pr(Y\in A\mid x_0\leq X\leq x_0+\epsilon) = \frac{\Pr(Y\in A, x_0\leq X\leq x_0+\epsilon)}{\Pr(x_0\leq X\le... | 531 |
probability distributions | What is the difference between dexp and pexp | https://stats.stackexchange.com/questions/111970/what-is-the-difference-between-dexp-and-pexp | <p>I understand probability distribution but I am having a hard time getting a grasp on probability density function, specifically difference between dexp (density of exponential distribution) and pexp (probability distribution of exponential distribution)</p>
| <p>Suppose X is an exponential random variable.
pexp(c) is the probability that X is less than or equal to c. pexp is always non-decreasing. To prove this, let m>0, then pexp(c+m)=P(X
<p>dexp(c) is the derivative of pexp(c), but intuitively, it is the probability that X is 'near' c, or the 'density' of the probabilit... | 532 |
probability distributions | joint probability distribution with a constant | https://stats.stackexchange.com/questions/124620/joint-probability-distribution-with-a-constant | <p>If $X \sim N(0,1)$, then what is the joint probability distribution of $(X+1,X)$?</p>
<p>An attempt: $f(x,x+1)=f(x|x+1)f(x+1)=f(x+1)$, so $N((0,0),(0,0;0,1))$. Note sure though...</p>
| <p>First, define $Y=X+1$ to make the notation easier. What you are looking for is the pdf of YX. Here it is:</p>
<p>$$ f_{YX}(y,x) = f_{Y|X}(y|x)f_X(x).$$</p>
<p>We know $f_X(x)$, so the question is what is $f_{Y|X}(y|x)$ ?</p>
<p>The answer is the following. If we know that $X=x$, then $Y=X+1 = x+1$. So $Y$ is a co... | 533 |
probability distributions | Comparing values of random variables from sampled distributions | https://stats.stackexchange.com/questions/128732/comparing-values-of-random-variables-from-sampled-distributions | <p>This is a novice question, which I struggling to answer. I want to find P(A>B) where A and B are random variables from two different distributions. Although two distributions that I have are not classical variants (e.g. normal or binomial etc) but a sampled distribution, I cannot use analytical approach where I coul... | 534 | |
probability distributions | rate of convergence of sample mean | https://stats.stackexchange.com/questions/134325/rate-of-convergence-of-sample-mean | <p>Law of large number ensures the convergence of sample mean to population mean. But it does not tell about the rate of convergence. Now CLT tells about the rate of convergence. Then what is the necessity of Berry-Essen theorem which also gives us the rate of convergence?</p>
| 535 | |
probability distributions | Distribution of the ratio of dependent magnitude square of complex Gaussians | https://stats.stackexchange.com/questions/137192/distribution-of-the-ratio-of-dependent-magnitude-square-of-complex-gaussians | <p>Assume that $X=X_1 + X_2 +...+X_n$, where $X_i \sim CN(0,\sigma^2)$ and independent. Here $CN$ means circular complex Gaussian.</p>
<p>The question is, what is the distribution for</p>
<p>$Z = \frac{\left|X\right|^2}{\left|X_1\right|^2 + \left|X_2\right|^2+...+\left|X_n\right|^2}$</p>
<p>How can we benefit from t... | 536 | |
probability distributions | Are first passage times in a Brownian motion the same as the ARL in a CUSUM? | https://stats.stackexchange.com/questions/32784/are-first-passage-times-in-a-brownian-motion-the-same-as-the-arl-in-a-cusum | <p>I am investigating a link between a random walk with drift (call it Brownian process or difusion with drift) and the CUSUM statistic.<br>
The CUSUM procedure accumulates deviations from the process mean over time, thus, if a change in the mean occurs for some reason, then the CUSUM will steadely increase over time, ... | <p>Brownian motion with drift is a linear function with Gaussian noise added to it. Generally speaking CUSUM charts are used to detect a process going out of control. So what is assumed is that the process starts out under control and something could go wrong. This would show up in a change in the mean of some measu... | 537 |
probability distributions | Fit a distribution to a combinatorial problem | https://stats.stackexchange.com/questions/34517/fit-a-distribution-to-a-combinatorial-problem | <p>In my previous question titled <a href="https://math.stackexchange.com/questions/183614/conditional-combinations-of-balls-in-bowls">"Conditional combinations of balls in bowls"</a>, is there a distribution to fit $k$ when $d \gg M$? I mean, when $d$ is so large, what is the distribution of the total number of balls?... | <p>Using uppercase $K$ for a random variable and lowercase $m,d,k_0$ for constants:</p>
<p>The number of balls in a particular bowl has a <a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28discrete%29" rel="nofollow">discrete uniform distribution</a> with mean $\frac{m}{2}$ and standard deviation $\sqrt{\fr... | 538 |
probability distributions | Does any one know what is the P value when F=37.45; df1=5; df2=40 in a one way anova F test? | https://stats.stackexchange.com/questions/41476/does-any-one-know-what-is-the-p-value-when-f-37-45-df1-5-df2-40-in-a-one-way-a | <p>in a one way anova F test
when F=37.45; df1=5; df2=40
what is the P value?
I tried several software, and the result is <0.0001. I know it sounds weird that I need a really small number of possibility. However, I really need it for a publication.
I greatly appreciate if anyone can help on this issue. Please let me... | <pre><code>0.00000000000004551914400963141815737
</code></pre>
<p>Is that exact enough?</p>
<p>Seriously, even though in a publication it asks for an exact amount there's a point where just saying it's less than 0.0001 is all you can reasonably do. It's a very very very small number. I strongly suggest that you not... | 539 |
probability distributions | Distributions similar to the family of stable distributions | https://stats.stackexchange.com/questions/49413/distributions-similar-to-the-family-of-stable-distributions | <p>Are there any other distributions with similar properties to the family of <a href="http://en.wikipedia.org/wiki/Stable_distributions" rel="nofollow">stable distributions</a>? That is, $\alpha$-stable, normal tempered stable, classical tempered stable, etc. etc. where the distribution could exhibit the following pro... | 540 | |
probability distributions | Probability distribution to simulate number of users | https://stats.stackexchange.com/questions/52987/probability-distribution-to-simulate-number-of-users | <p>I need to simulate with probability distribution number of users who are listening radio during the day. Given data : max number of users is 1000, day has 24 hours (from 0 to 24), the highest number of users should be from 11 to 15. Which probability distribution will be the best and which parameters of distribution... | <p>You may use a <a href="http://en.wikipedia.org/wiki/Birth%E2%80%93death_process" rel="nofollow">Birth-Death</a> process: a "birth" is a new user tuning in, a "death" is a listener tuning out. If you adjust the rates of the process properly, you will not need to introduce the maximum users cut-off explicitly.</p>
| 541 |
probability distributions | Probability density function scalar-valued non-linear function of continuous random variables | https://stats.stackexchange.com/questions/546723/probability-density-function-scalar-valued-non-linear-function-of-continuous-ran | <p>Is it possible to numerically calculate the probability density function for a scalar-valued function where each variable is independent and has a known distribution? For example in the simple case of <span class="math-container">$z=f(x,y)$</span>:
<span class="math-container">$$
z=y*e^x
$$</span>
if <span class="ma... | 542 | |
probability distributions | Probability distributions in an ordered set of random extracted elements | https://stats.stackexchange.com/questions/548420/probability-distributions-in-an-ordered-set-of-random-extracted-elements | <p>I tried looking for an existing solution for the following problem:</p>
<ul>
<li>Assume that S is a set of d elements, and R is a total order relation on S. Assume that n elements are randomly extracted from S, and then they are ordered according to R. Which is the probability that in the i-th position of the ordere... | 543 | |
probability distributions | When do I have 80% chance of 6 rolls of either 1, 2 or 3? | https://stats.stackexchange.com/questions/550125/when-do-i-have-80-chance-of-6-rolls-of-either-1-2-or-3 | <p>Problem:
Every roll has 4 outcomes: 1, 2, 3 and 4
There are 50% chance for 1-3 at every roll. (And that means 50% for 4)
If one roll did not give 1-3 at one roll, then it is 100% chance at the roll after. (So every second roll is guaranteed to be either 1, 2 or 3)
If rolled 1, 2 or 3 then there is equal chance of ei... | <p>To take <span class="math-container">$r$</span> rolls, you have to get</p>
<ul>
<li><span class="math-container">$6$</span> rolls of one of the three numbers <span class="math-container">$1,2,3$</span>,</li>
<li>and <span class="math-container">$a$</span> rolls of the next with <span class="math-container">$0\le a \... | 544 |
probability distributions | Why do we need weighted distributions? | https://stats.stackexchange.com/questions/550520/why-do-we-need-weighted-distributions | <p>I have read some papers about Weighted Distribution. Suppose <span class="math-container">$X$</span> is a non-negative continuous random variable with pdf <span class="math-container">$f(x)$</span>. The pdf of the weighted random variable <span class="math-container">$X_w$</span> is given by:</p>
<p><span class="mat... | <p>Situations in which weighted distributions occur or have some use:</p>
<ol>
<li><p>Mixture models of the type <span class="math-container">$f(x)=\sum_{k=1}^K \pi_kf_k(x)$</span>. To clarify: not the mixture itself is a weighted distribution, rather a mixture component <span class="math-container">$f_k$</span> is <sp... | 545 |
probability distributions | Distribution of number of balls in a subset of bins | https://stats.stackexchange.com/questions/289244/distribution-of-number-of-balls-in-a-subset-of-bins | <p>I have $ B $ bins and $ G $ balls, and each of the balls has a weight. I toss the balls uniformly into the bins. I select the $ K $ bins that have the highest weights as determined by the weights of the balls that are in them, and I then record the total number of balls that appear in these $ K $ bins. </p>
<p>I wa... | 546 | |
probability distributions | What is the probability that the best N people come from China? | https://stats.stackexchange.com/questions/494227/what-is-the-probability-that-the-best-n-people-come-from-china | <p>Consider two countries competing in a game like chess. And suppose the abilities of all individuals have distributed according to Uniform[0, 1] distribution.</p>
<p>Say country <code>A</code> has population <span class="math-container">$P$</span> and country <code>B</code> has population <span class="math-container"... | <p>Thanks to uniform randomness, that would be simply <span class="math-container">$$P(N>k)=P(N\geq k+1)=\frac{{P \choose k+1}}{P+Q\choose k+1}$$</span></p>
<p>It's choosing best <span class="math-container">$k+1$</span> people from population of <span class="math-container">$P$</span> people vs all possible choices... | 547 |
probability distributions | Support of a probability distribution | https://stats.stackexchange.com/questions/555657/support-of-a-probability-distribution | <p>Consider a bivariate probability distribution <span class="math-container">$G$</span> and a random vector <span class="math-container">$(X_1,X_2)$</span>. Should <span class="math-container">$G$</span> satisfy any specific <strong>support</strong> restrictions in order to be an admissible probability distribution ... | 548 | |
probability distributions | Probability finishing on a given turn two independent draw | https://stats.stackexchange.com/questions/557482/probability-finishing-on-a-given-turn-two-independent-draw | <p>I have two decks of card. I draw from each deck independently. I need to find a specific card from each individual deck. What is the odd that on turn <span class="math-container">$n$</span>, I have found the card in each deck (this is not an assignment)?</p>
<p>From what I understand, the probability to finish on th... | <p>I am not quite sure what your <span class="math-container">$P(2)$</span> is supposed to be,</p>
<p>but if it is the probability that you have seen both individual cards by the second turn drawing from both decks but not both by the first turn and you are drawing without replacement,</p>
<p>then it should be <span cl... | 549 |
probability distributions | pdf of the product of two independent uniform random variables $X,Y \sim U(-1,1)$ | https://stats.stackexchange.com/questions/560253/pdf-of-the-product-of-two-independent-uniform-random-variables-x-y-sim-u-1-1 | <p>Using the product distribution. I have <span class="math-container">$Z = XY$</span> with <span class="math-container">$X,Y \sim U(-1,1)$</span> and independent. Thus
<span class="math-container">\begin{align}
f_Z(z) & = \frac{1}{4} \int_{-1}^1 I_{[-1 < \frac{z}{x} < 1]} \ \frac{dx}{|x|} \\
& = \frac{1}... | 550 | |
probability distributions | Are $U + V$ and $UV$ independent when $U,V$ are independent and standard uniform? | https://stats.stackexchange.com/questions/560260/are-u-v-and-uv-independent-when-u-v-are-independent-and-standard-uniform | <p>This is related a previous question I posted on the product of two independent variables <a href="https://stats.stackexchange.com/questions/560253/pdf-of-the-product-of-two-independent-uniform-random-variables-x-y-sim-u-1-1">here</a>. As an alternative method, one could note that if <span class="math-container">$X,Y... | <p><strong>Here's a very simple solution.</strong> It involves no integration and only the easiest algebra.</p>
<p>Let <span class="math-container">$X=2U-1$</span> and <span class="math-container">$Y=2V-1.$</span> These are <em>iid</em> uniform random variables on <span class="math-container">$[-1,1]$</span> and ther... | 551 |
probability distributions | Probability distribution of questions in a forum | https://stats.stackexchange.com/questions/11749/probability-distribution-of-questions-in-a-forum | <p>I would like simulate appearance of publications in a forum and I need know what is the probability distribution of new question being asked in a forum. In my first simulation I used to normal distribution, but I think that the best distribution can be exponential distribution.</p>
| <p>The exponential distribution might be a good starting point for the <em>waiting time</em> between new posts. This would be equivalent to assuming a Poisson distributed number of posts in a given time period. There are some pretty strong assumptions behind a model like that, but it might make sense for your applicati... | 552 |
probability distributions | If a sequence of distributions converges to a degenerate, does that imply the variance strictly decreases? | https://stats.stackexchange.com/questions/16872/if-a-sequence-of-distributions-converges-to-a-degenerate-does-that-imply-the-va | <p>If $F_i = G(F_{i-1}), F_0 = x$ is a sequence of distributions that converges to a degenerate distribution as $i \to \infty$, does that imply that the variance of $F_i$ decreases with $i$? </p>
<p>Specifically, I am interested in the inverse Kumaraswamy distribution: $G(x) = (1-(1-x)^a)^b$ when $a=b$... Note that $... | <p><strong>Here is a simple counterexample.</strong></p>
<p>Assume the CDF $F$ is supported on $[0,1]$ and define a new CDF $G[F]$ as follows. If $\mathbb{E}[F] \gt 1/2$, let</p>
<p>$$G[F](x) = 1 - F(3/2 - 2 x).$$</p>
<p>Otherwise, let</p>
<p>$$G[F](x) = 1 - F(5/6 - 2x/3).$$</p>
<p>The first operation squeezes th... | 553 |
probability distributions | A recurrence relation in an elementary problem in probability? | https://stats.stackexchange.com/questions/489852/a-recurrence-relation-in-an-elementary-problem-in-probability | <p>While considering solving the following standard question in probability in an alternative way, I get stuck with a recurrence relation. The problem is as follows:</p>
<blockquote>
<p>A bag contains <span class="math-container">$N$</span> ropes. We pick up two randomly chosen ends and tie them together until no untie... | 554 | |
probability distributions | Joint distribution given normalized gamma distributed components | https://stats.stackexchange.com/questions/497368/joint-distribution-given-normalized-gamma-distributed-components | <p>Consider <span class="math-container">$N = 2^{n}$</span> random variables <span class="math-container">$X_{1}, X_{2}, \ldots, X_{N}$</span>, such that for each <span class="math-container">$i \in [N]$</span>,</p>
<p><span class="math-container">$$X_{i} \sim \Gamma\left(\frac{1}{2}, 2^{-n+1}\right). $$</span></p>
<p>... | 555 | |
probability distributions | Basic question about PDFs: multi-parameter PDF, not multi-variate | https://stats.stackexchange.com/questions/501787/basic-question-about-pdfs-multi-parameter-pdf-not-multi-variate | <p>I'm kinda new to stats and trying to find info on PDFs that depend on multiple parameters, but I keep finding info only on multi-variate distributions.</p>
<p>The point is that <strong>I have only one random variable</strong> <span class="math-container">$R$</span>, but it depends on two or more parameters, say <spa... | <p>If you only have the <code>age</code> and <code>weight</code> density, you cannot do much else than multiply them, and then you end up with the same <code>age</code> distribution for every <code>weight</code> for example, which is probably not what you are looking for.</p>
<p>The problem is that how <code>weight</co... | 556 |
probability distributions | how many times will get exactly two heads if I toss 2 not-fair coins (chances of getting a head is 0.68) for 100 times? | https://stats.stackexchange.com/questions/502338/how-many-times-will-get-exactly-two-heads-if-i-toss-2-not-fair-coins-chances-of | <p>I want to calculate how many times I will get exactly two heads if I toss 2 not-fair coins (chances of getting a head is 0.68) for 100 times?</p>
<p>Can somebody gives me a formula that I can generalize this question? (n not-fair coins where is p the chance of head for m times)</p>
| <p>This is just a binomial distribution where “success” is getting two heads. Let’s calculate the probability of such an event.</p>
<p><span class="math-container">$$P(HH) = P(H)P(H)=0.4624$$</span></p>
<p>So this is our “p” in the binomial distribution. The other parameter in the binomial distribution is <span class="... | 557 |
probability distributions | Cumulative distribution function equals almost surely | https://stats.stackexchange.com/questions/506072/cumulative-distribution-function-equals-almost-surely | <p>Let <span class="math-container">$F_1, F_2$</span> - two continuous CDF.</p>
<p>if <span class="math-container">$F_1 = F_2\quad F_2$</span> almost surely (i.e. probability of <span class="math-container">$x$</span> where <span class="math-container">$F_1(x)\neq F_2(x)$</span> is zero with respect to probability with... | <p>By contrapositive, if exists <span class="math-container">$x$</span> such that <span class="math-container">$F_1(x) \neq F_2(x)$</span></p>
<p>if <span class="math-container">$F_2(x) < F_1(x)$</span>, then choose <span class="math-container">$y>x$</span> such that <span class="math-container">$F_2(y) > F_2(... | 558 |
probability distributions | Probability of trial sequence | https://stats.stackexchange.com/questions/508660/probability-of-trial-sequence | <p>A trader learns to predict whether the stock price will rise or fall on a particular day of trading. To do this, he calls one hundred friends and asks them to toss a coin once a day, thus receiving one hundred signs of the type "heads / tails from the n-th friend." What is the probability that within a wee... | <p><strong>Yes, Bernoulli distributions are involved.</strong> But they are combined in a complex way; and it is important to be clear about how you understand the question. Here is one proposal.</p>
<h3>Framing the problem</h3>
<p><strong>Let us say that "100% correlates with the dynamics of stocks" means ... | 559 |
probability distributions | trials from a multinomial distribution required to obtain 1 item in particular | https://stats.stackexchange.com/questions/269611/trials-from-a-multinomial-distribution-required-to-obtain-1-item-in-particular | <p>I have a set of outcomes $\{A, 2A, B, C\}$, which on a roll appear with probabilities $\{0.10, 0.15, 0.40, 0.35\}$ respectively.</p>
<p>How many rolls do I need to get 30 $A$ elements?</p>
| 560 | |
probability distributions | Adjust probabilities to make it equal to 1 | https://stats.stackexchange.com/questions/273673/adjust-probabilities-to-make-it-equal-to-1 | <p>Is there a way to normalize the given probabilities? Let's say for example I have this probability for the following words:</p>
<pre><code> I am the best today
</code></pre>
<p>$$1/5 + 1/5 + 1/5 + 1/5 + 1/5 = 1$$</p>
<p>Given that people typed the word</p>
<pre><code>am
</... | <p>As I understand you correctlly, you want "am" to have 0.5 probability, keep relations between other probabilities and make all probabilities sum to 1. Am I right?</p>
<p>If so, you should first normalize the four probabilities you want to normalize to make them sum to 1. You can do this by dividing them by their su... | 561 |
probability distributions | Probable value of a | https://stats.stackexchange.com/questions/274005/probable-value-of-a | <p>I'm working on writing an excel program that logs a series of measurements. The measurements are then used in 6 different models, all approximating the same unknown value.</p>
<p>I'd like to take the result of these 6 models and approximate the most probable value for what the unknown is. I've used average and medi... | <p>If you have several estimators of the same value, you can combine them by using a weighted average of all the estimates where the weights are given based on the variance of each estimator. Check for example: <a href="https://en.m.wikipedia.org/wiki/Inverse-variance_weighting" rel="nofollow noreferrer">https://en.m... | 562 |
probability distributions | Is the following inequality correct? How to prove it? | https://stats.stackexchange.com/questions/274566/is-the-following-inequality-correct-how-to-prove-it | <p>Suppose $X$ and $Y$ are two arbitrary random variables, and we have the following inequality that conditional on $Y=y$,
$$\textbf{Pr}(X \ge a_0 | Y=y)\le f(y),$$
where $\textbf{Pr}(\cdot)$ denotes the probability of the event, $a_0$ is a constant, and $f(y)$ is an increasing function with respect to $y$. I want to ... | <p>Expand using product rule:</p>
<p>$$P(X \ge a_0 \cap Y \le b_0) = P(X \ge a_0 | Y \le b_0) P(Y \le b_0)$$</p>
<p>Assuming $P(Y \le b_0)$ is nonzero, you can use the first inequality to obtain:</p>
<p>$$P(X \ge a_0 | Y \le b_0) P(Y \le b_0) \le f(y\le b_0) P(Y \le b_0)$$</p>
<p>Because $f(y)$ is increasing in $y$... | 563 |
probability distributions | What is meant by dynamic range when talking about probability distribution? | https://stats.stackexchange.com/questions/278299/what-is-meant-by-dynamic-range-when-talking-about-probability-distribution | <p>To provide some context, the exact sentence is</p>
<blockquote>
<p>for right-skewed data of the kind we consider here the method is especially sensitive to slight deviations of the data from the power-law model around xmin because most of the data, and hence most of the dynamic range of the CDF, lie in this regio... | 564 | |
probability distributions | A distribution on binary vectors defined by a sum | https://stats.stackexchange.com/questions/296643/a-distribution-on-binary-vectors-defined-by-a-sum | <p>The following distribution came up in my research and I'm wondering if the general class has a name and if there are computationally efficient sampling methods for it.</p>
<p>Let $\mathcal{X} = \{0,1\}^{n}$ be the sample space; these are simply $n$ dimensional vectors with each component either $0$ or $1$. Let $a_1... | <p>It's multinomial distribution. Same as repeating a coin toss n times and count the number of times head appears. In this case you also have coefficient to each i th toss ( i= 1..n) means that you have treat ith toss differently from the i+1 th toss. </p>
| 565 |
probability distributions | Probability of false negative in uniform distribution test | https://stats.stackexchange.com/questions/287874/probability-of-false-negative-in-uniform-distribution-test | <p>Let's say I have a set of $n$ objects, and I select an object from the set with uniform probability. I do this many times, and record how many times I select each object. These counts will tend toward a uniform distribution, but an exact uniform distribution is obviously unlikely. </p>
<p>What's the chance that the... | 566 | |
probability distributions | Given two means and deviations, how can I compute the probability that x < y? | https://stats.stackexchange.com/questions/297927/given-two-means-and-deviations-how-can-i-compute-the-probability-that-x-y | <p>I've experimentally achieved my goal by running many random trials, generating two points according to scaled and translated gaussian distributions and counting how many times x is less than y.</p>
<p>However this is becoming a problem from a performance point of view for my application.</p>
<p>Is there a straight... | <p>If your assumptions regarding the normality of $X$ and $Y$ are correct, then you have two normally distributed random variables $X\sim\mathcal{N}(\mu_X, \sigma_X^2)$ and $Y\sim\mathcal{N}(\mu_Y, \sigma_Y^2)$, and you are searching for $\mathrm{Pr}(X < Y)$, aka $\mathrm{Pr}(X-Y < 0)$.</p>
<p>Assuming $X$ and $... | 567 |
probability distributions | Calculate probability of number based on previous numbers in a non-random sequence | https://stats.stackexchange.com/questions/300718/calculate-probability-of-number-based-on-previous-numbers-in-a-non-random-sequen | <p>I have a large number of number sequences of different lengths consisting of 1s and 0s indicating whether an action has occurred, like:</p>
<pre><code>0111001111101
00010001111
01111111011111
</code></pre>
<p>The digits are not independent, as they represent whether an action has been performed or not, which is ... | 568 | |
probability distributions | Taking expectations over uniformly distributed random variable | https://stats.stackexchange.com/questions/303134/taking-expectations-over-uniformly-distributed-random-variable | <p>I have a probably very silly question, but somehow I am lost and don't get to the solution. Either it is my mistake or there is a mistake in the paper. Hopefully you can help me out!</p>
<p>The problem is as follows: The density of random variable q that is uniformly distributed between 0 and $\bar{q}$ is given by:... | 569 | |
probability distributions | Distribution of X³/Y where X and Y are uniformly distributed | https://stats.stackexchange.com/questions/298607/distribution-of-x%c2%b3-y-where-x-and-y-are-uniformly-distributed | <p>$X$ and $Y$ are two uniformly distributed variables in interval $(a,b)$ and $(c,d)$ respectively (a>0 and c>0). </p>
<p>What is the distribution of the variable $Z$ defined as $Z=X^3/Y$?</p>
| 570 | |
probability distributions | Distribution of a random variable to the nth power | https://stats.stackexchange.com/questions/302187/distribution-of-a-random-variable-to-the-nth-power | <p>If we know the distribution of $X$ is symmetric, do we know anything about the shape of the distribution of $X^n$, where $n\geq 0$ but is not necessarily an integer?</p>
<p>I'm not referring to the moments of $X$, but rather the distribution of its products (but more generally, raised to the $n$th power).</p>
<p>E... | 571 | |
probability distributions | Probability distribution for random walk samples of Voronoi decomposition of high-dimensional space of exponentially-distributed points | https://stats.stackexchange.com/questions/309510/probability-distribution-for-random-walk-samples-of-voronoi-decomposition-of-hig | <p>I've got a protein I'm modeling. One thing I do with it is randomly perturb the protein for a fixed amount of time at a chosen temperature, and then minimize it so that its configuration reaches a local energy minimum. Higher temperatures can reach energy minima further and further away from the starting configurati... | 572 | |
probability distributions | Independent joint conditional probability | https://stats.stackexchange.com/questions/315227/independent-joint-conditional-probability | <p>I am new to conditional probability. I would like to do some inference. If we know p(z|x) and p(z|y), can we infer p(z|x,y)? What can we deduce if x and y are independent? Thank you very much.</p>
| <p>There is not enough information in $p(z|x)$ and $p(z|x)$ to determine $p(z|x,y)$. In particular, when $X$ and $Y$ are independent, they may become dependent conditional on $Z$: take for instance the exponential variates $X$, $Y$, and $Z$, related by
$$X\sim\mathcal{E}(1)\qquad Y\sim\mathcal{E}(1)\qquad Z|X,Y\sim\mat... | 573 |
probability distributions | Is $Pr(x \leq C)$ equal to $Pr(\sqrt{x} \leq \sqrt{C})$? | https://stats.stackexchange.com/questions/321935/is-prx-leq-c-equal-to-pr-sqrtx-leq-sqrtc | <p>Where x is non-negative continuous random variable and C is a constant.</p>
| <p>Yes, because $x \leq C \Leftrightarrow \sqrt{x} \leq \sqrt{C}$ for $x, C \geq 0$. </p>
<p>This means that the set of events $\{A \in \Omega: x(A) \leq C\}$ equals the set $\{A \in \Omega: \sqrt{x(A)} \leq \sqrt{C}\}$, so are their probabilities.</p>
| 574 |
probability distributions | On upper bounds on the distance between the population density and a chosen density | https://stats.stackexchange.com/questions/324440/on-upper-bounds-on-the-distance-between-the-population-density-and-a-chosen-dens | <p>Say One has a certain number of observations from a population density $p(x)$ and based on the observations decides to utilize a density $q(x)$ to approximate the unknown population density $p(x)$.</p>
<p>Can one choose $q(x)$ in such a way to get some upper bound for the $L^2$ distance or $L^1$ distance between th... | 575 | |
probability distributions | Probability distribution, n dice | https://stats.stackexchange.com/questions/330508/probability-distribution-n-dice | <p>Suppose I have $n$ fair dice with 6 faces each (numbers from $1$ to $6$). I define a random variable $X_n$ - sum of numbers on dice after the throw. Is there any probability distribution I can use to calculate exact probabilities for every value of this random variable? I'm interested in a distribution that gives ex... | 576 | |
probability distributions | How to compute hypergeometric distribution probabilities for complex events? | https://stats.stackexchange.com/questions/350461/how-to-compute-hypergeometric-distribution-probabilities-for-complex-events | <p>How would I calculate the sums of two (or more) hypergeometric distributions. If, using a standard deck of cards, I want to determine the probability of draw 2 red cards and one Black Queen. I cannot just change my "good card" size and use one formula cause that wouldn't tell me what I need.</p>
<p>So, given 52 car... | <p>Since drawing more red cards implies you have drawn fewer black cards, the red card count and black queen count are not independent. That makes it difficult to combine the probabilities of each event in any simple way to obtain the answer.</p>
<p>Instead, do it the old-fashioned way: count every hand having two re... | 577 |
probability distributions | probability of repeated events | https://stats.stackexchange.com/questions/371379/probability-of-repeated-events | <p>I have a website and I want to calculate the probability of clicks on the ads.</p>
<p>Let the probability that each user clicks on a link be <code>p</code> (something like <code>1%</code>)</p>
<p>if we have totally <code>N</code> users, What is the formula that computes the probability of exactly <code>n</code> c... | <p>Consider each user as a trial. For every trial you have two outcomes, they are success (clicks on ad) and failure (does not click on ad). <span class="math-container">$P[success] = p $</span> and <span class="math-container">$P[failure] = 1-p$</span>. </p>
<p>The total number of ways in which <span class="math-cont... | 578 |
probability distributions | Probabilities under the log normal distribution, as well as mean and sd | https://stats.stackexchange.com/questions/372729/probabilities-under-the-log-normal-distribution-as-well-as-mean-and-sd | <p>I have what are probably a pretty basic stats questions. I heard that under the log normal distribution, the mean =variance. Is this true? Or is this another distribution? I am having trouble finding this information online.</p>
<p>Second, if I have the mean and sd for a log normal distribution, how to I calculate ... | <p>"I heard that under the log normal distribution, the mean =variance. Is this true?" No. For normal distribution, mean and variance have no relation. For lognormal distribution, mean and variance have relation, but not determined each other.</p>
<p>"Or is this another distribution?" One of them is Poisson distributi... | 579 |
probability distributions | pdf of combined models | https://stats.stackexchange.com/questions/376201/pdf-of-combined-models | <p>In a question, I have 5 systems. At a given time x, the probability that they're working is based on an exponential distribution. The combination of all systems will work if any single system is working</p>
<ul>
<li>Systems fail independently</li>
</ul>
<p>How can I determine the pdf of the system reliability? (ie... | <p>Call <span class="math-container">$T_i$</span> as the lifetime of system <span class="math-container">$i$</span>; then, your total system lifetime, <span class="math-container">$T=\max(T_1..T_5)$</span>. You'll first write <span class="math-container">$F_T(t)=P(T\leq t)=\prod_{i=1}^5 P(T_i\leq t)$</span>, and differ... | 580 |
probability distributions | When drawing three number from the same distribution, what is the probability of the first to be between the two? | https://stats.stackexchange.com/questions/383688/when-drawing-three-number-from-the-same-distribution-what-is-the-probability-of | <p>If I draw 3 numbers: <span class="math-container">$a$</span>, <span class="math-container">$b$</span> and <span class="math-container">$c$</span> from the exact same distribution (unknown, but the same for each of the numbers).
I want to know the probability that <span class="math-container">$a$</span> is between <s... | <p>Any of the three can be the one in the middle. If you think it may be more probable that <span class="math-container">$a$</span> is the one in the middle than <span class="math-container">$b,$</span> then what happens if you rename them so that the one called <span class="math-container">$a$</span> is then called <s... | 581 |
probability distributions | Calculate $P(X>10)$ where $X$ have Poisson distribution | https://stats.stackexchange.com/questions/384461/calculate-px10-where-x-have-poisson-distribution | <p>Calculate <span class="math-container">$P(X>10)$</span> where <span class="math-container">$X$</span> have Poisson distribution <span class="math-container">$Poisson(7,2)$</span>. using <span class="math-container">$R$</span></p>
<p><b>My attempt</b>
By the theory of probability we know <span class="math-contain... | <p>I think that qpois’s output is a quantile, ppois gives a probability.</p>
<p>Seems you’re looking for a probability, so the first formula is OK.</p>
<p>For the second question, you might use ppois(8,lambda) - ppois(1,lambda) where lambda is the parameter of your variable.</p>
| 582 |
probability distributions | How to find the value for average rate | https://stats.stackexchange.com/questions/387084/how-to-find-the-value-for-average-rate | <p>I'm doing some textbook problems on my own and there are no steps given to the solutions to some of the problems. If anyone could help me solve the following problem, much would be appreciated.</p>
<p>"A liquid culture medium contains on the average <em>m</em> bacteria per ml. A large number of samples is taken, ea... | <p>Since you figured it out, I will post the answer. Poisson is the easiest option, as you figured out, since it has the least number of parameters. So using the Poisson formula</p>
<p><span class="math-container">$$P(k \leq 0)=P(k=0)=0.1$$</span></p>
<p>Which means</p>
<p><span class="math-container">$$e^{-\lambda}... | 583 |
probability distributions | How can I calculate the probability of a increasingly likely positive outcome? | https://stats.stackexchange.com/questions/388611/how-can-i-calculate-the-probability-of-a-increasingly-likely-positive-outcome | <p>I'm not even sure I'm phrasing the question properly, please let me know if there is any standard terms around this type of problem.</p>
<p>I'm trying to find an average number of attempts it would take to find a randomly placed pixel within a fixed size element. If a wrong pixel is chosen, it is removed from the c... | <p>You're looking to calculate expected value of a variable <span class="math-container">$X$</span>, which we define as number of clicks until success.</p>
<p><span class="math-container">$$E[X] = \sum_{i=1}^{10,000} i \cdot Pr(X = i)$$</span> </p>
<p>Note that for each possible <span class="math-container">$i \in \{... | 584 |
probability distributions | Probability Theory and distribution | https://stats.stackexchange.com/questions/394331/probability-theory-and-distribution | <p>I am studying probability theory and one of the questions that I have faced is this. The problem is that I either don't know where to go about with this question or even if I do do something about it, I have no way of knowing if I'm on the right path. Is it possible to please help explain this?</p>
<p>Q. In Germany... | <blockquote>
<p>In Germany, every year 125 people out of 100 000 report the occurrence of Parkinson's Disease. a) Which kind of distribution would you use to model the occurrence of new cases per year?</p>
</blockquote>
<p>This can be modeled as <a href="https://en.wikipedia.org/wiki/Poisson_distribution" rel="nofol... | 585 |
probability distributions | What happens if my Anderson Darling score is more than 10? | https://stats.stackexchange.com/questions/394819/what-happens-if-my-anderson-darling-score-is-more-than-10 | <p>I am using crystal ball to fit a distribution curves. Based on the fitting, the AD score is 20, p value is 0. What does it mean? Does it mean it is suitable?</p>
<p>Appreciate any help!</p>
<p>Thanks!</p>
| <p>With an exceptionally high AD score of 20, and a p value that is probably equal to 0.0000% you can reject the null hypothesis that your data may have a distribution that could come from the same population as the distribution you are testing for. </p>
<p>I suspect you used the AD test to check if your data is norm... | 586 |
probability distributions | Biased coins: Probability such that the first player throws heads | https://stats.stackexchange.com/questions/403974/biased-coins-probability-such-that-the-first-player-throws-heads | <p>Let <span class="math-container">$A$</span>, <span class="math-container">$B$</span> be the two players. Each one has a coin has a probability of getting heads of <span class="math-container">$p_i$</span>. Player <span class="math-container">$A$</span> always starts first. What is the probability such that <span cla... | <p>If you drawn a tree, then you can see that either <span class="math-container">$A$</span> wins straight away, or <span class="math-container">$A$</span> flips tail and <span class="math-container">$B$</span> also flip tail and then <span class="math-container">$A$</span> gets heads, ..., or <span class="math-contain... | 587 |
probability distributions | Accounting for membership in set, but also quantity in event | https://stats.stackexchange.com/questions/403084/accounting-for-membership-in-set-but-also-quantity-in-event | <p>Say we have a sample space
<span class="math-container">$$\Omega_1 = \{\text{"alpha"},\text{"beta"},\text{"gamma"},\text{"delta"}\}$$</span>
if we only care about the (binary) membership in set in each event, the event space would be the power set of <span class="math-container">$\Omega_1$</span>:
<span class="math... | <p>What you want here can be accomplished by using the sample space:</p>
<p><span class="math-container">$$\Omega = \{ 0,1 \}^4 \times \mathbb{R}^4.$$</span></p>
<p>This sample space allows for four binary indicators and four corresponding real values. The event space <span class="math-container">$\mathscr{G}$</span... | 588 |
probability distributions | how to obtain the distribution of random variables? | https://stats.stackexchange.com/questions/416234/how-to-obtain-the-distribution-of-random-variables | <p>If we have two independent random variables modeled as triangular distribution centered around a mean in the interval (mean - a, mean + a). 'a' is finite value. Then discretized them at fixed sub-intervals. what will be the distribution of their max?</p>
| <p>Your question can be answered by considering the question: "in what ways can the <span class="math-container">$k$</span> be the maximum of these two random variables?"</p>
<p>Case 1: <span class="math-container">$X < k$</span> and <span class="math-container">$Y = k$</span>. This happens with probability <span c... | 589 |
probability distributions | Is a transformation of a conditional distribution identical to the conditional of the transformation? | https://stats.stackexchange.com/questions/407932/is-a-transformation-of-a-conditional-distribution-identical-to-the-conditional-o | <h1>Problem</h1>
<p>Say that we can find a random vector <span class="math-container">$(Y_1, \cdots, Y_n)$</span> whose distribution is identical to the conditional distribution of <span class="math-container">$(X_1, \cdots, X_n)$</span> under <span class="math-container">$f(X_1, \cdots, X_n) \le 1$</span>, where <spa... | 590 | |
probability distributions | Total variation of a distribution | https://stats.stackexchange.com/questions/413757/total-variation-of-a-distribution | <p>The wikipedia page for <a href="https://en.wikipedia.org/wiki/Total_variation#Total_variation_of_probability_measures" rel="nofollow noreferrer">Total Variation</a> says that "The total variation of any probability measure is exactly one" (and is therefore not interesting).</p>
<p>I don't get why. </p>
<p>For exam... | <p>If you use the definition <span class="math-container">$\Delta = \sup_i \sum_j |\mu(E_j^i)|$</span> you get exactly one. In fact for your example, let's <span class="math-container">$E = \{0, 1, 2, 3\}$</span> and <span class="math-container">$E_i:=\{E^i_j\}$</span> a part of <span class="math-container">$E$</span>.... | 591 |
probability distributions | For any distributions, are mean of the sum of the same distributions equals the sum of the means of the distributions? | https://stats.stackexchange.com/questions/420123/for-any-distributions-are-mean-of-the-sum-of-the-same-distributions-equals-the | <p>I have some distributions with the same distribution, for example, Gaussian distribution or Beta distribution. My question: is the mean of the sum of these distributions equals to the sum of the means of these distributions?</p>
| <p>Yes. Mostly. By the property called linearity of expectation, the mean of the sum of random variables is the same as the sum of their individual means. </p>
<p>This is true so long as there are finitely many random variables going into the sum and so long as their means are finite. Also true in some other cases, se... | 592 |
probability distributions | An example of a random variable whose both marginal distributions and itself has the same probability distribution? | https://stats.stackexchange.com/questions/421577/an-example-of-a-random-variable-whose-both-marginal-distributions-and-itself-has | <p>To ask what I want to, let me start with an example:</p>
<p>Let <span class="math-container">$\chi(x)$</span> and <span class="math-container">$\eta(t)$</span> be two independent random variables. Then we can define a new random variables such as
<span class="math-container">$$\epsilon_1(x,t) = \chi(x) + \eta(t),$$... | 593 | |
probability distributions | How probability distributions help a statistical analyst/data scientist | https://stats.stackexchange.com/questions/431238/how-probability-distributions-help-a-statistical-analyst-data-scientist | <p>How exactly the probability distributions help a statistician/data scientist in modelling/decision making? </p>
<p>Or how using distributions a data scientist derive any inference or make decisions when modelling? </p>
<p>I understand probability distributions theoretically but not sure how its utilised in day to ... | <p>I suppose you are referring to the standard probability distributions? In that case, an example would be the arrival times for a certain service. </p>
<p>For example, suppose that a hospital has a limited amount of beds to accommodate patients. The hospital wants to know how many beds it should have at any given ti... | 594 |
probability distributions | Drawing curve from events that have probabilities | https://stats.stackexchange.com/questions/430480/drawing-curve-from-events-that-have-probabilities | <p>I'm having some problems plotting my data, and understanding what I can do with it</p>
<p>I have a dataset that looks something like:</p>
<pre><code>| Event | Score | Prob |
+-------+-------+------+
| 1 | 5 | 30% |
| 2 | 2 | 90% |
| 3 | 1 | 20% |
| 4 | 9 | 30% |
| ... | ... | <p>You could work out the mean score by multiplying each score by its probability, and summing.</p>
<p>This might not be the most likely (mode) score if the distribution is not symmetrical, or if it is multi-modal.</p>
<p>You could run a simulation to get this plot (assuming the events are independent):
Iterate 10000... | 595 |
probability distributions | Is it appropriate use the Binomial Theorem to analyze the problem of rolling dice? | https://stats.stackexchange.com/questions/431476/is-it-appropriate-use-the-binomial-theorem-to-analyze-the-problem-of-rolling-dic | <p>In mathematics, the <a href="https://en.wikipedia.org/wiki/Multinomial_theorem" rel="nofollow noreferrer">multinomial theorem</a> </p>
<blockquote>
<p>describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomia... | <p>This really depends on exactly what you are looking at. "Rolling dice" is a specification of <em>an activity</em>, not a specification of a <em>numerical outcome</em> that constitutes a random variable having a distribution. If you roll a set of standard six-sided dice and you look at the counts of the six possibl... | 596 |
probability distributions | Distribution of the top $m$ of $n$ samples of a Gaussian distribution? | https://stats.stackexchange.com/questions/435061/distribution-of-the-top-m-of-n-samples-of-a-gaussian-distribution | <p>I was wondering if there was an analytic description of the distribution of the largest <span class="math-container">$m$</span> of <span class="math-container">$n$</span> samples of a Gaussian distribution, where <span class="math-container">$n \geq m$</span>.</p>
<p>(As an example, I generated 100 samples from <sp... | 597 | |
probability distributions | Distribution of uniform RVs under sum constraint | https://stats.stackexchange.com/questions/438090/distribution-of-uniform-rvs-under-sum-constraint | <p>Suppose I generate <span class="math-container">$x_1,x_2,x_3,x_4$</span> through the following procedure:</p>
<ol>
<li>Sample <span class="math-container">$x_1,x_2,x_3 \sim \text{unif}(0, 1)$</span>, iid </li>
<li>While <span class="math-container">$x_1+x_2+x_3 > 1$</span>, resample them all</li>
<li>Let <span ... | <p>The answer is:that's not true.
it is look like </p>
<ol>
<li>generate <span class="math-container">$(X_1,X_2,X_3)$</span> such that <span class="math-container">$S=X_1+X_2+X_3 \leq 1$</span></li>
<li><span class="math-container">$X_4=(1-S)$</span></li>
</ol>
<p>or on the other hands <span class="math-container">... | 598 |
probability distributions | If odd is uniformly distributed, what is the distribution of proportion? | https://stats.stackexchange.com/questions/448872/if-odd-is-uniformly-distributed-what-is-the-distribution-of-proportion | <p>Suppose <span class="math-container">$\pi=\frac{\theta}{1-\theta}$</span> where theta is between <span class="math-container">$[0,1]$</span>. </p>
<p>If we set a uniform prior for <span class="math-container">$\pi$</span> (<span class="math-container">$p(\pi) \propto 1$</span>), what is the induced prior on <span c... | <p>If <span class="math-container">$\pi$</span> is uniformly distributed over <span class="math-container">$(0,a)$</span> then <span class="math-container">$\theta\in\left(0,\frac{a}{1+a}\right)$</span>. Then for <span class="math-container">$0<x<\frac{a}{1+a}$</span>
<span class="math-container">$$
F_\theta(x)=\... | 599 |
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