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.

6,104 questions with no upvoted or accepted answers
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22
votes
0answers
2k views

A difficult integral

For $\gamma>0,\delta>0$, trying to evaluate this integral: $$ I=\int_0^H\frac{e^{i t x} \log\left(\frac{H}{H-x}\right) ^{\frac{1}{\gamma }-1} \left(\left(\frac{k}{H \log \left(\frac{H}{H-x}\...
14
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0answers
213 views

Brownian motion and Beta distribution

I am interested in the distribution of the time that the standard Brownian $W_t$ motion on $[0,1]$ satisfies the following inequality: $$W_t \ge stW(1)$$ For different values of $s$. I conjecture that ...
11
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2answers
148 views

Intuitive explanation of CDF of a Binomial distribution in the volume of a Hyperspherical Cap

Note: This is my first question ever in stackexchange, I apologize for any mistakes in formatting, on the appropriateness of the question and tags. From Wikipedia, I know the regularized incomplete ...
11
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0answers
407 views

Fabius function and equivalent

The Fabius function $F$ can be defined on $[0,1]$ by $F(0)=0$ $F(1)=1$ on $[0,\frac{1}{2}]$ $F'(x)=2.F(2x)$ on $[\frac{1}{2},1]$ $F'(x)=2.F(2(1-x))$ It's a known example of a not analytic $C^\...
11
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0answers
459 views

Is this question solvable? $2$ non-linear equations and the proof that the solution is unique

As mentioned in the title I want to show the uniqueness of the solution to $2$ non-linear equations. However, it seems that I can not solve this question with my current mathematical knowledge. More ...
11
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0answers
374 views

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 ...
10
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1answer
133 views

Constructing a probability measure on the Hypercube with given moments

Let $H = [-1, 1]^d$ be the $d$-dimensional hypercube, and let $\mu \in \text{int} H$. Under these conditions, I can explicitly construct a tractable probability measure $P$, supported on on $H$, ...
10
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0answers
1k views

Difficult integral for a marginal distribution

I am trying to derive a marginal probability distribution for $y$, and failed, having tried all methods to solve the following integral: $$p(y)=\int_0^{\frac{1}{\sqrt{2 \pi }}} \frac{\sqrt{\frac{2}{\...
10
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0answers
753 views

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|$....
10
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1answer
2k views

Singular jacobian matrix?

I have a series of questions, in various degrees of befuddled muddledness (and they are related to my previous questions: this and this) First question: how do I do a change of variable if the ...
9
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0answers
734 views

the parametrization of a Gumbel in terms of a Gaussian

Extreme Value Distribution From a Gaussian. I was wondering how the parametrization of $\alpha$ and $\beta$ of a Gumbel $e^{-e^{-\frac{x-\alpha }{\beta }}}$ was done in terms of a cumulative Gaussian $...
9
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0answers
1k views

What is the distribution of $\sqrt{X^2+Y^2}$ when $X$ and $Y$ are Gaussian but correlated?

If $Z = \sqrt{X^2+Y^2}$, and $X$ and $Y$ are zero-mean i.i.d. normally-distributed random variables, then $Z$ is Rayleigh distributed. What is the distribution of $Z$ if $X$ and $Y$ are correlated (...
8
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0answers
171 views

Is it true that $\phi(\mu)=\mu F(\mu)^2-\int_{\mu}^{\overline{v}}F(v)[1-F(v)]dv\geq 0$?

Consider a random variable $V$ with distribution function $F$ and density function $f$ with support $[\underline{v},\overline{v}]$, where $0\leq\underline{v}<\overline{v}$. The mean is $\mu$. Here $...
8
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0answers
86 views

Optimal rate of growth of i.i.d. Gaussians?

Suppose I have a countable collection $\{N_k\}_{k=1}^\infty$ of independent $\mathcal{N}(0.1)$ random variables and I want to estimate their rate of growth in the sense that I want to find a function $...
8
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0answers
181 views

Transformations of RV's Ensuring Absolute Continuity of Quantile Functions

Given a real random variable $X$, suppose $T:\mathbb{R}\to\mathbb{R}$ is non-decreasing. Define $Y=T\left(X\right)$. Let $Q_{X}$, $Q_{Y}$ be the corresponding right-continuous quantile functions. ...
8
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0answers
574 views

Random matrices, eigenvalue distribution.

I just investigated randn(1024) + 1i*randn(1024), a 1024x1024 complex valued matrix with elements both real part and imaginary part drawn from $\mathcal{N}(\mu = 0, \sigma = 1)$. I was a bit surprised ...
8
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0answers
1k views

What is a spherical Gaussian kernel?

In this paper (page 8, Example 3), a spherical Gaussian kernel is defined by the formula $$K(\mathrm x,\mathrm y)=e^{-2\epsilon(1- \mathrm x\cdot\mathrm y)}$$ where $\mathrm{x,y}\in S^{n-1}\subseteq\...
8
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0answers
446 views

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 ...
7
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0answers
105 views

Enhanced Berry-Esseen theorem for the digits of $\sqrt{2}$

The Berry-Esseen theorem provides a second-order approximation to the central limit theorem (itself a first order approximation.) Higher order approximations are available, see here. If the random ...
7
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0answers
178 views

Only three types of limit of distributions truncated to a finite interval in the upper tail?

Suppose random variable $X$ has a continuous probability distribution with an unbounded upper tail; that is, the CDF of $X$ (call it $F$) is absolutely continuous and $F(x)<1$ for all $x\in\mathbb{...
7
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0answers
138 views

Getting a bound for Gibbs distribution mean

Suppose $F$ is a strictly convex and increasing function, $U$ a random variable with support $[0,1]$ and density $$ f_U(u)= \frac{e^{-\frac{1}{T}F(u)}}{\int_{0}^{1} e^{-\frac{1}{T} F(x)} dx}.$$ Do we ...
7
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0answers
380 views

Regular Version of Conditional Gaussian Distribution

Let $Z_{1}$ and $Z_{2}$ be two independent normally distributed random variables with expectations $\mu_{1},\mu_{2}\in\mathbb{R}$ and variances $\sigma_{1}^2,\sigma_{2}^2\in (0,\infty)$ . I would ...
7
votes
1answer
659 views

Simulating from a Multivariate Gaussian without Cholesky

I'd like to draw a sample from a multivariate Gaussian distribution $\mathcal{N}(\mu, \Sigma)$, where $\mu$ is the mean vector (can assume it to be $\boldsymbol{0}$), and $\Sigma$ is a sparse positive ...
6
votes
1answer
165 views

Finding Probability that Maximum Value of $N + 1$ Random Variables is Larger than a Given Limit

For positive random variables $X_{0}, X_{1}, X_{2}, \ldots, X_{N}$, where $N$ is also a random variable, we know that $X_{1}, X_{2}, \ldots, X_{N - 1}$ are I.I.D. with continuous PDF $f(x)$ and $X_{0} ...
6
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0answers
480 views

How to use Kullback-Leibler Divergence if probability distributions have different support?

I have two discrete random variables $X$ and $Y$ and their distributions have different support. Assume $X$ and $Y$ can both take on the same number of values. Lets say $X$ takes values in $\{10,13,15,...
6
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0answers
196 views

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 ...
6
votes
1answer
480 views

markov chain: 2 state chain

I have a machine. It has two states, broken or working. If it is working, then it will be broken with probability $q=0.1$. If the machine is working, I will make \$1000 dollar a day. If it is broken, ...
6
votes
2answers
154 views

Estimate Grade Distribution Based on Performance of Each Question

As the title states, I would like to be able to estimate the grade distribution of an exam based on the mark distribution of each individual question. To give a quick example of what I mean, suppose ...
6
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0answers
210 views

What is the Lévy measure of the Student's $t$-distribution?

It is known since the 1970's that the Student's $t$-distribution is infinitely divisible. We can therefore apply the Lévy-Khintchine representation to it, and define the Lévy measure associated to a ...
6
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0answers
152 views

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 $X$ ...
6
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0answers
149 views

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 $\exp(\lambda_2)$....
6
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0answers
250 views

Distance between a Poisson and Normal distribution.

Let $X_a$ be a random variable Poisson distributed with intensity $a$. That is $$\mathbb{P}(X_a=k)= e^{-a} a^k / (k!)$$ for any $k\in \mathbb{N}$. Let $$Y_a=(X-a)/\sqrt{a}$$ the normalization of $...
6
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0answers
218 views

How to compute or simplify this integration?

Any hints on solving an integration of the following form, $$\int_{x}^{+\infty}\left(1-\frac{1}{1+sy^{-1}}\right) \left(\text{exp}(-\sqrt{y})+ y^{-\frac{1}{2}}(1-\text{exp}(-\sqrt[4]y)\right)dy $$ ...
6
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0answers
231 views

Expected area of an inscribed triangle in a sphere

On the surface of a unit sphere, three points $A$, $B$ and $C$ are chosen in the following way: Points $A$ and $B$ are chosen randomly and independently on the whole surface After $A$ and $...
6
votes
1answer
480 views

Understanding the setup for the probability that $Ax^2+Bx+C$ has real roots if A, B, and C are random variables uniformly distributed over (0,1).

Suppose that $A, B,$ and $C$ are independent random variables, each being uniformly distributed over $(0,1)$. What is the probability that $Ax^2 + Bx + C$ has real roots? First, I set $P(B^2 - 4AC \...
6
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0answers
559 views

Gamma random variables with fixed sum (different scale parameters)

Given a vector of independent random variables $\{X_i\}_{i=1..N}$, each of which is distributed according to a Gamma-distribution with pdf $Pr(X_i=x;\alpha_i,\beta_i) = \frac{1}{\Gamma (\alpha_i)}\...
6
votes
0answers
850 views

Maximum and minimum of an integral under integral constraints.

Find the maximum and minimum of the following integral in terms of $f(x),a,C$: \begin{align}I=\int_{0}^{a} \frac{x}{f(x)}p(x)dx \end{align} s.t.: 1) $\int_{0}^{a} p(x)dx=1$ 2) $\int_{0}^{a} f(x)p(x)...
6
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0answers
265 views

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 ...
5
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0answers
159 views

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 $X$ ...
5
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0answers
108 views

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 ...
5
votes
1answer
185 views

approximation of product of two independent Normal random variables

I know that the product of two independent zero mean Normal random variables follows the normal product distribution. I want to talk about the random variable $$z = xy$$ where $x \sim \mathcal{N}(0,1)...
5
votes
0answers
85 views

What is the distribution of the difference of two normalized binomial random variables?

Let $X \sim Bin(n, p)$ and $Y \sim Bin(m, p)$. How is $$Z_1 = \frac{X}{n} - \frac{Y}{m}$$ and $$Z_2 = \left|\frac{X}{n} - \frac{Y}{m}\right|$$ distributed? (Hence: What is their cumulative ...
5
votes
0answers
2k views

Jacobian Transformation to find joint pdf of $Y_1$ and $Y_2$

Let $X_1$ and $X_2$ have the following joint density: $$f_{X_{1},X_{2}}(x_1,x_2) = \begin{cases} {\frac{4}{\pi}e^{-(x_1^2+x_2^2)}} & \text{$x_1 \gt 0$, $x_2 \gt 0$} \\{0} & \text{...
5
votes
0answers
505 views

How to get joint probability density from bivariate distribution function

Let $X_{i} \sim \varepsilon(\lambda_{i}), i = 1,2,3$ be mutually independent ($\varepsilon$ means exponential, $\lambda_{i}$'s are parameters). Then $(T_{1},T_{2}) = (X_{1} \wedge X_{3}, X_{2} \wedge ...
5
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0answers
82 views

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 $\Omega$....
5
votes
0answers
195 views

Expectation of $\frac{X_{1}+X_{2}+X_{3}+…+X_{k}}{X_{1}+X_{2}+X_{3}+…+X_{n}}$ , $1 \leq k \leq n$?

Prove the following statement: Let $X_{1},X_{2}, \dots ,X_{n}$ be a set of exchangeable random variables. Then, $$E\left(\frac{X_{1}+X_{2}+X_{3}+\dots +X_{k}}{X_{1}+X_{2}+X_{3}+\dots +X_{n}}\right) ...
5
votes
0answers
114 views

Definition of expectation value in quantum mechanics

I've read the following proposition in a book on quantum theory. Proposition. If a quantum system is in a state described by a unit vector $\psi$ and for some quantum observable $\hat{f}$ we have ...
5
votes
0answers
228 views

PDF of difference of two i.i.d. random variables: maximum at $0$ and decreasing to the right of $0$?

Let $X, Y$ be two i.i.d. random variables with an arbitrary distribution. As discussed here, the distribution of their difference $X-Y$ is symmetric around $0$. What I am wondering: Does this ...
5
votes
0answers
77 views

Expectation of increasing transformation of random variables

Suppose that $X$ is a positive continuous random variable with infinitely differentiable pdf $f_\theta (x)$ and suppose that its expectation is increasing in $\theta$. That is, the function $$ g_{1}(\...
5
votes
0answers
217 views

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 ...