# Questions tagged [probability-distributions]

Questions on using, finding, or otherwise relating to probability distributions, probability density functions (pdfs), cumulative distribution functions (cdfs), or other related functions.

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### Distribution of the sum of absolutes values of T-distributed random variables

Where X is a r.v. following a symmetric T distribution with 0 mean and tail parameter $\alpha$. I am looking for the distribution of the n-summed independent variables $\sum_{1 \leq i \leq n}|x_i|$....
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### Do probability distributions form a comonad?

$\def\unit{{\rm unit}}\def\join{{\rm join}}$It's well known that (discrete) probability distributions form a monad. Specifically, if we let $PX$ be the set of discrete probability distributions on ...
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### Calculating probability of some event using geometric considerations

I want to estimate exponentially the following probability: Let $\bf{U}\in\mathbb{R}^n$ be a random vector uniformly distributed on the $n$-dimensional hypersphere, centered at the origin with radius ...
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### Is this fraction undefined? Infinite Probability Question.

Where $\frac{1}{\infty}$ and $\frac{\infty}{\infty}$ are both undefined terms that generally lead to nonsense, I'm wondering if we can assert that...: $$\frac{1+1+1+\cdots}{1+1+1+\cdots} = 1$$ ...or ...
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### Uniqueness of the transformation turning random variables into IID uniform

We have two random variable $X:\Omega \to \mathbb R$ and $Y: \Omega \to \mathbb R^d, d \in \mathbb N$, $F_Y$ is the density function of $Y$ and $F_{X|Y=y}$ is a regular density function of $X$ ...
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### Conditional expectation involving some complications around exponential random variables

Here is my problem. Consider four independent exponential distributions $X^A_1$, $X^B_1$, $X^A_2$, $X^B_2$ where $X^A_1$ and $X^B_1$ are $\exp(\lambda_1)$ and $X^A_2$ and $X^B_2$ are $\exp(\lambda_2)$....
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### I need help about some compactness arguments

I need help to find some compact sets for my engineering problem. Through this page I learned quite much about it however since I have neither read a book yet nor have an experience I am not able to ...
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### Finding the support of the CDF of $(X,Y)$

Assume $X$ to be standard normal random variable, and define $Y$ as$$Y=\begin{cases}X,&\text{if }⌊X⌋\text{ is even}\\-X,&\text{if }⌊X⌋\text{ is odd}\end{cases}.$$ I am trying to show that $X$ ...
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### Expected value of the sample median from a folded normal distribution

Suppose $X_1, \ldots, X_n \sim N(0,\sigma^2)$ are iid. Find the expected value of $M$, the median of $\vert X_1 \vert, \ldots \vert X_n \vert$ What I have so far: The density of $\vert X_i \vert$ is ...
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### Quantiles of comonotone sums

Let $(\Omega, \mathcal F, P)$ be a probability space. Let $\mathbf{X} = (X_1, X_2, \ldots, X_n)^T$ be a random vector and $U \sim \mathrm{uniform}(0, 1)$ be a random variable, both defined on $\Omega$....
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### Randomly Generate Probability Mass Function With Specific Entropy

How can I randomly generate a probability mass function such that the entropy of a random variable that follows that probability mass function is a specific value $h$? Basically, I need to randomly ...
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### Maximum Elo Rating.

I'm trying to implement a variant of the Elo system, for a game I'm working on. Giving two players $A$ and $B$ with ratings $R_A$ and $R_B$ respectively, the expectation of $A; E_A$ is given by the ...
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### Entropy of the multinomial distribution

What is the entropy of the multinomial distribution? To fix notation, let us define $n > 0$ as the number of trials, $p_1, \ldots, p_k$ as the probabilities of each of the $k$ possible outcomes and ...
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### Suppose $E[X_1] <\infty$. Show that $\lim_{n\rightarrow \infty} \frac{X_n}{S_n}=0$ a.s.

Let $X_1,X_2,X_3,...$ be i.i.d. with $P(X_1 >0)=1$. Define $S_n =\Sigma_{i=1}^{n} X_i$. (a) Suppose $\mathbb{E}[X_1] <\infty$. Show that $\lim_{n\rightarrow \infty} \frac{X_n}{S_n}=0$ a.s. I ...
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### Distribution with many cumulants vanishing

Let $X$ be a random variable. It is well-known that $X$ is normally distributed if and only if its last cumulants $\kappa_3 = \kappa_4 = ... = 0$ vanish. I was wondering if there are standard ...
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### Compound Distribution — Normal Distribution with Normally Distributed Mean

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Normal Distribution whose mean is distributed Normally. ...
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### Symmetric proof for the probability of real roots of a quadratic with exponentially distributed parameters

What is the probability that the polynomial has real roots? asked for the probability that the quadratic polynomial $ax^2+bx+c$ has real roots if the parameters $a,b,c$ are exponentially distributed ...
The normalizing factor for the product of two multivariate Gaussian densities, $f(x)$ and $g(x)$ with mean vectors $a$ and $b$ respectively, and covariance matrices $A$ and $B$ respectively, is itself ...