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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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Statistics and Probability Expected Value Questions

Compute the frst two moments (that is EY and EY^2) of Y = 2^-X if X = b(n, p) (binominal distribudion). Random variable X has the distribution P(X = k) = c*3^-k(keN). Determine the value of c. Does ...
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Modified prior distribution

Suppose we have a prior $p(z)$ and conditional likelihood $p(x|z)$ whose integral gives the evidence $p(x)=\int p(x|z)p(z)dz$. Now let $q(x)$ be a distribution different from $p(x)$ and let $p_q(z)=\...
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Covariance between squared bivariate geometrically distributed random variables

If $X$ and $Y$ follows bivariate geometric distribution (where $EX=a$, $EY=b$, $Cov(X,Y)=c$ ) then how to obtain (determine) $Cov(X^2,Y)$?
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Probability of random variable given probability of its modulus

Assume that $z$ is a random vector. Also, assume that $$z = A(\|y\|)$$ where A is whitening function and $\|\cdot\|$ is complex modulus (norm) such that if $$y_j = a_j + ib_j,$$ then $$\|y_j\|=\...
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Discrete random variables are equal

Let's say we want to find the distribution of $Y$, such that for any suitable function $g$, random variables $g(X)$ and $Y$ have the same distribution. Also, $X,Y$ are continuous. We could solve $$E(\...
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Naming a restricted probability space

Consider a probability space $(\Omega,F,P)$. I want to consider $(\Omega',F',P')$, where $\Omega'\subset\Omega$ is nonempty event such that $P(\Omega')>0$, and $F'=\{\omega\cap\Omega'\mid\omega\in ...
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35 views

Inequalities for standardized central moments of probability distributions

It's known that the standardized central even moments of any probability distribution with a density symmetric around the mean form a non-decreasing series, the lower bound (when all are equal to 1) ...
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How to compute the optimal values of $\lambda_1,\ldots,\lambda_K$?

So, here's a question that has come up in my research work. Suppose $P_1$ and $P_2$ are $M\times M$ transition probability matrices such that $P_1\neq P_2$. Further, let $\mu_1$ and $\mu_2$ denote the ...
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1answer
45 views

Distribution of $\frac{X^{t}AX}{X^{t}X}$ in multivariate normal distribution

In the multivariate normal case for $X\sim N_{p}(0,I)$, consider an idempotent matrix $A$ of rank $k<p$. I need to find the distribution of $\frac{X^{t}AX}{X^{t}X}$. I know that $X^{t}AX$ follows ...
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1answer
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Express $P\{\{X_1+X_2\leq \alpha \}\cap\{X_3\leq X_2\}\}$ using CDF and PDF

Let $X_1$ $X_2$ and $X_3$ be three positive independent random variables. The PDF and CDF of $X_i$ are denoted by $f_{X_i}(x_i)$ and $F_{X_i}(x_i)$ respectively. I would like to compute the ...
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693 views

Maximum Likelihood Estimate with Multiple Parameters

I am not very familiar with multivariable calculus, but something tells me that I don't need to be in order to solve this problem; take a look: Suppose that $X_1,...,X_m$ and $Y_1,...,Y_n$ are ...
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Bartlett`s approximation for Wilks Lambda

For two given independent wishart distributed variables A and B, where $$A\sim W(\Sigma,m),~B\sim W(\Sigma,n)$$ it implies for the transformed variable $$\lambda=\frac{det(A)}{det(A)+det(B)}\sim \...
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What is the large-n limit of a distribution of the following sample statistic:

What is the large-n limit of a distribution of the following sample statistic:$$\displaystyle\frac{\sum^{n}X_{i}}{\,\sqrt{\,\sum^{n}X_{i}^{2}\,}\,}$$ when sampling the Cauchy(0,1) distribution? Monte ...
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1answer
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Show $\lim \left| \left( 1-(1-s)\frac{\lambda_n}{n}\right)^n-\left(1-(1-s)\frac{\lambda}{n}\right)^n\right|\le\lim|1-s ||\lambda_n-\lambda |$

As application of convergence theorem in our probability lecture we want to show the generating function of sequence of binomially distributed random variables converges to the generating function of ...
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Sequence of samplings from a fixed population

I have a population of size $N=100$ of which $f=33$ of them are bad. I'd like to create $k=5$ subpopulations of size $m=20$ out of the given population that satisfies the following conditions. ...
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The distribution function is $F(x)=\begin{cases}1-e^{-x}, & \text{$x$>0} \\ 0, & \text{$x$≤0 } \end{cases}$ But I don't know why

Let the random variables $X$ and $Y$ be independent,$X$ follows the exponential distribution with parameter $1$, and the probability distribution of $Y$ is $$\mathcal{P}\{Y=-1\}=p,\;\mathcal{P}\{Y=1\}...
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1answer
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What sampling distribution does $\hat{p}$ in this scenario follow?

Homework question, not sure whether I should be using a normal distribution, a binomial distribution or a hypergeometric distribution. A mail-order company promises its customers that the products ...
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the probability distribution of Y is P{Y=-1}=p P{Y=1}=1-p,(0<p<1),set Z=XY. [on hold]

Let the random variables $X$ and $Y$ be independent,$X$ follows the exponential distribution with parameter $1$, and the probability distribution of $Y$ is $P${Y=$-1$}=$p$ $P${Y=$1$}=$1$-$p$,set$ Z=XY$...
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Finding $c$ such that $ \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} ce^{-(x^2 - xy + 4y^2)/2}\,dx\,dy = 1 $

Let $f(x, y)$ be the given function. $$f(x, y) = ce^{-(x^2 - xy + 4y^2)/2}$$ Determine the value of $c$ such that $f$ is a pdf. The problem is finding the value of $c$ such that $$ \int_{-\infty}...
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Given $E[X] , \operatorname{Var}(X)$ and $Y\mid X \sim U(X,1)$, find $E[Y]$ and $\operatorname{Var}(Y)$

For $X, Y$ random variables, given $E[X] = \mu$ ; $\operatorname{Var}(X) = \sigma^2$; $Y\mid X \sim \text{Unif}(X,1)$: Find $E[Y]$ and $\operatorname{Var}(Y)$. (1) To find E[Y], I used the law of ...
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renewal equation for $\sim U(0,1)$ interarrivals

renewal equation for $\sim U(0,1)$ interarrivals should be $m(t)=t+\int_0^t{m(t-s)f(s)ds}$ how can this be solved? can I make substitution $y=t-s$ to get $m(t)=t+\int_0^t{m(y)f(y)dy}$ if all I ...
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2answers
120 views

Joint distribution of consecutive renewal times

Consider a discrete analog to the Poisson process. Let the sequence $X_i$ be independent geometrically (with parameter $p$) distributed random variables that signify the inter arrival times of events. ...
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166 views

renewal process question

The question and answer below is related to the renewal process. I'm curious how the "I" changed to "H(T)" as indicated by the yellow boxes. Thanks for spending the time to look over my newbie ...
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1answer
44 views

Distribution of a discontinuous random variable and it's convergence in the law

I'll start this question with a two definitions that are given in the Chapter 1 of the book: Da Prato, Zabczyk - Stochastic equations in infinite dimensions, 1992. If $(\Omega,\mathcal{F})$ and $(E,\...
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1answer
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Find $P\{\{X_1+X_2\leq \alpha\}\cap\{X_1\leq X_2\}\}$ for $X_1$ and $X_2$ have the same parameter or different parameter

Let $X_1$ be the maximum of $N_1$ iid exponential random variables with parameter $\beta_1$. Similar, let $X_2$ be the maximum of $N_2$ iid exponential random variables with parameter $\beta_2$. The ...
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Approximate the CDF of the sum of 2 probability distributions

Given two complicated CDFs $f(x)$ and $g(x)$ on random variables $X$ and $Y$ respectively. Is there a way to approximate for the combined CDF $h(x)$ on $X + Y$ without having to perform a convolution ...
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Symmetricity in Kaiming initialization

Kaiming initialization paper says that "If we let $w_{l-1}$ have a symmetric distribution around zero and $b_{l-1 } = 0$, then $y_{l-1}$ has zero mean and has a symmetric distribution around zero". ...
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1answer
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Function returning a random variable whose probability decreases with its value.

For my programming project I need a function that returns a random variable X of type double in range [1, +infinity>. This variable should follow a simple rule: the higher the value, the lower the ...
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Probability that $AX^2+BX+C=0$ has only real roots for $A,B,C \sim Unif(0,1)$ [duplicate]

From chapter 6, practice exercise 26 (b) from "A First Course in Probability" by Sheldon Ross (working it for my own recreation): Given R.V.s $A,B,C\sim_{iid}Unif(0,1)$ and asked to find the ...
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Probability Distributions (Tree Diagram)

Satish picks a card at random from an ordinary pack. If the card is ace, he stops; if not, he continues to pick cards at random, without replacement, until either an ace is picked, or four cards have ...
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Square of uniform distribution over offset Interval

Let $X\sim U\left[-2, 1\right]$. I want to calculate $F_{X^2}(t)$, but I don't know how to handle the interval. With, e.g., $\left[-2, 2\right]$, it would be trivial: $ F_{X^2}(t) = \mathbb{P}\...
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Distribution of $\sum_k \alpha^k X_k$

Let $(X_k)$ be a sequence of independent Bernoulli random variables, such that $\Pr[X_k = 1] = p$. Then for $0\le\alpha<1$ the sum $$\sum_{k=0}^\infty \alpha^k X_k$$ is real random variable in the ...
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subtracting winning probabilities

Two players play a game of chess. When they get home, they use a computer to analyze the game. For each position of the game, the computer knows what move is best, and it assigns a winning ...
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What is an appropriate and general function signature associated with a probability distribution?

Let $X$ be a random variable with probability distribution $f_X$ Then $f_X$ is a function, $f_X : A \to B, c \mapsto d$ What exactly are $A,B,c,d$? I am sorry if this question is basic, as I ...
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967 views

Beta distribution times a scalar

If I have a random variable that has a Beta distribution multiplied by a scalar (say 1000), what is its distribution then? I have been doing some research and it appears not to be a beta distribution. ...
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High probability bound for the The Double Dixie Cup Problem

The Double Dixie Cup Problem is similar to the Coupon collector's problem where a collector buys a random card out of $n$ at a time, but its goal is to collect $m$ copies of each card (the Coupon ...
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How do we find $M$ which maximizes the expression $\lVert f_{n}(x,M)\rVert_{2}$?

Let $p:[0,1]\rightarrow\mathbb{R}_{\geq0}$ be a continuous function whose integral over the interval $[0,1]$ equals one. Consider as well the (unknown) continuous and bounded function $M:[0,1]\times\...
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Change double integral into two convolutions

I'm looking to change a double integral into convolution from the original integral, which is $$S(z_0,\Omega) = \int_{4\pi}\frac{H\!\left(z_0,\Omega,\Omega'\right)}{4\pi}\int_{z_0}^{\infty}F\!\left(z'...
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Incoming call in a business processing unit… [on hold]

a) Incoming calls are monitored in a Business processing unit. Call arrival is modeled as Poisson distribution with an average rate of $18$ per hour. Determine the probability that the time interval ...
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Overlap lower bound between distributions

Suppose that I have multinomial distribution with probabilities $p_i$, for $i \in [1,n]$. I'd like to find a lower bound for the sum of the overlapping probabilities between $p_i$'s and the ...
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Poisson-Approximation of Poisson-Binomial distribution

I'm looking for a proof as to why the Poisson-Binomial distribution can be approximated by the Poisson distribution. A Poisson-Binomial distribution is a sum of independent Bernoulli trials that are ...
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Rate of convergence of (random) minimum Euclidean distance

Suppose two random variables $X$ and $Y$ are independent and both distributed uniformly on $[0,1]$. I am interested in the vectors $Z=(X, Y)$. Suppose I have $N$ independent realizations of $Z$: $Z_{...
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Invariant distribution of jump chain in terms of invariant distribution of corresponding continuous Markov process

Let $Q$ be the Q-matrix of a continuous time Markov process ($q_{ii}=-\sum\limits_{j\ne i}q_{ij}$ and $q_{ij}\ge 0\ \forall i\ne j)$ The probability matrix of the jump chain corresponding to the ...
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What is the expected length of the smaller piece?

A stick of length $1$ is broken into two pieces by cutting at a randomly chosen point. What is the expected length of the smaller piece? (a)$1 /8$ (b)$1 /4$ (c)$1 /e$ (d)$1 /π$ This is a question ...
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Finding distribution of $y$ intercept given by line generated by two uniformly chosen points in $[0,1]^2$

The method I'm using seems super inefficient. What I did was define $4$ RVs namely $X_1,X_2,Y_1,Y_2$ and thus my two uniformly random points are $(X_1,Y_1),(X_2,Y_2)$ and hence the $y$ intercept is $-...
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1answer
44 views

Posterior of a corrupted transformation

Let $z\sim\mathcal{N}(z|0,1)$ and let $p(x|z)=\mathcal{N}(x|f(z),\gamma)$ where $f:\mathbb{R}\to\mathbb{R}$ is a differentiable bijection. I am trying to approximate the posterior $p(z|x)$ with a ...
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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 ...
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What is considered to be a Heap’s law?

I’m not sure if this is more physics question than mathematics but anyways. Something is usually said to follow Heap’s law if it is given as a function $V(n)=K n^b$, where $b$ and $K$ are constants (...
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Is $\Pr[X \leq x] \leq \max(\Pr[X + Y \leq x], \Pr[X - Y\leq x])$

Suppose that $X$ and $Y$ are random variables which can be arbitrarily dependent. Does the following inequality hold: \begin{equation} \Pr[X \leq x] \leq \max(\Pr[X + Y \leq x], \Pr[X - Y\leq x]). \...
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Uniqueness of 1-dimensional distributions via uniqueness of Laplace Transform

Consider a stochastic process $X = (X_t)_{t \geq 0 }$ on $\mathbb{R}$ and two probability measures $\mathbb{P}_1,\mathbb{P}_2$ with expectations denoted $\mathbb{E}_1,\mathbb{E}_2$. Suppose for $g$ ...