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Mathematics Stack Exchange

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. Use this tag along with the tags (probability), (probability-theory) or (statistics).

Any probability distribution, including beta, binomial, chi, Erlang, gamma, geometric, lognormal, negative binomial, normal (Gaussian), Pareto, Poisson, Student's t, uniform, Wald, Weibull, zeta, and Zipf.

28080 questions
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I like to prove sum of new discrete random variable`s pmf is equal 1

First of all from last question I can obtain some discrete random variable which described below. In this page, we denote this random variable as $Z$. $P(Z=0) = e^{-\lambda}+(1-e^{-\lambda})(1-\beta)$ $P(Z=n\beta) =…
Kim
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Let $X\sim\text{unif} [0,1]$. Find the probability density function of $Y=(1/X)-X$.

Let $X\sim\text{Uniform} [0,1]$ and let $$Y=\frac{1}{X}-X.$$ Find the pdf of $Y$. $f(x)=1$ if $0≤x≤1$, $0$ otherwise $Y$ is a decreasing function. Thus, the pdf should be: $$f_y(y)=-f_x(g^-1(y)).|\frac{d}{dy}(g^-1(y))|$$ But how to find:…
A600
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Random Variable with p.d.f. as product of two $1d$ Gaussians?

$X_1 \sim N(\mu_1,\sigma^{2}_1)$ $X_2 \sim N(\mu_2,\sigma^{2}_2)$ What random variable as function of $X_1$ and $X_2$ has p.d.f. which is equal to product of two Gaussians?
Konstantin Burlachenko
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Which distribution is fully decided by the first four moments?

For example, Gaussian distribution is fully decided by the first two moments, i.e., mean and variance. Another example is the generalized Gaussian distribution which has one additional parameter, i.e., shaping parameter p, to control the fourth…
kawofengche
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Truncated Gumbel I extreme value probability density function (PDF)

Can anyone confirm the method is correct that I am using to get the truncated Gumbel I extreme value PDF. I am fitting Gumbel I to ice loads that occur on ships, if that is of any help here. The basic distribution functions. Gumbel I CDF of the…
wattaw
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What is the probability distribution for guessing the order of 6 known cards ( A-6)?

I have 6 cards on the table, Ace through 6. They are randomly placed face down in a row. I try to guess the correct order of the cards. It is possible to have zero correct guesses, or one correct, etc. I would like to know the correct distribution…
user306996
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Finding the joint PDF of a coordinate X of a point on the unit disk and its distance to the origin

I'm trying to solve the following question: Let $(X,Y)$ be a randomly chosen point on the unit disk, and let $R={(X^2+Y^2)}^{\frac12}$. Find the joint density of the vector $(X,R)$. I'm not sure whether my solution is correct. First, I've…
Brassican
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How to find a probability distribution of a variable given a PDF

If i have the region: $$W=\left\{ \left( x,y\right) \in \mathbb{R} ^{2}:0\leq y\leq x\leq 1\right\} $$ And the random vector $(X,Y)$ with the following joint PDF: $$ \begin{equation} f_{(X,Y)}(x,y)= \left\{ \begin{array}{@{}ll@{}} c(x^4 +…
Gotey
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The probability of sample covariance matrix to be positive definite is 1.

In the book "An introduction to multivariate statistical analysis" by T.W. Anderson, I note that the probability of $A=\sum_{i=1}^{N}(x_i-\bar{x})(x_i-\bar{x})^T$ with $\bar{x}=\sum_{i=1}^{N}x_i/N$ to be positive definite is $1$. How to prove…
shawn Wong
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Find the joint probability distribution function

I have a random vector $(X,Y,Z,W)$ with the following Probability Mass Function: The joint probability marginal distribution of $(X,Y)$ in a similar table with its own marginals is: $$(0,0)=0.225$$ $$(1,0)=0.2$$ $$(0,1)=0.275$$ $$(1,1)=0.3$$ How…
Gotey
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Decomposition of a truncated distribution

This question is actually a follow up to a question I asked earlier here, but I'll supply here all the necessary details. Let $X$ be a continuous random variable (rv) equipped with the CDF $$F_X(x)= \begin{cases} \frac12(2-e^{-x}) & x\ge0 \\…
Brassican
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A Question on Random Variable Distribution

Suppose the p.d.f of the random variable X is as follows: $$ f(x) = \left\{ \begin{array}{ll} 3x^2 & \mbox{for } 0< x <1 \\ 0 & \mbox{otherwise } \end{array} \right. $$ Also suppose that $$ Y = 1 - X^2 $$ What is the p.d.f. of Y? I need a…
AKH
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Bivariate Normal distribution problem.

The exponent $e$ of a bivariate normal density is given as follows : $$ -\dfrac{1}{102} [ (x+2)^{2} - 2.8(x+2)(y-1) + 4 (y-1)^{2} ]$$ We need to find : E(X) , E(Y) , Var(X) , Var(y) and $\rho$ (correlation coefficient). I tried to compare that…
User9523
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Marginal density function for dependent support

Given the joint density function of $(X,Y)$ is $$f(x,y)=\begin{cases} \frac{1}{8}(y^2-x^2)e^{-y} \quad \text{ if } -y
Jacky
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Bivariate Distribution

I am trying to understand bivariate probability distribution functions and I am following all of Statistics: a concise course in statistical inference book. In this book the author give one example for joint mass function as mentioned…
Venkatesh
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