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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. 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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Closed form for CDF of Beta distribution on $[a,b]$?

I am trying to derive the CDF of a Beta distribution defined on the interval $[a,b]$. I managed to get the PDF for this more general version of Beta, with the following result: $$ f(\alpha, \beta, x) = \frac{(x-a)^{\alpha-1}…
Paul
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Joint distribution of two random variables conditionally to a given $x$

Suppose that ($X$, $Y$ and $Z$) are three random variables with joint pdf \begin{align} f(x,y,z)=\begin{cases} \frac{1}{\pi}\exp(x(y+z-x-2))-\frac{1}{2}(y^2+x^2), & x\geq 0,\; x\in R, \; y\in R \\ 0, & \text{oterwise} \end{cases} \end{align} Ι have…
Spy93
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Integral of a power of a normal distribution

I have to solve the integral $\int_{-\infty}^{\infty} N(x; \mu, \sigma)^p\,dx$. I remember I had to do this before and came up with an easy and elegant solution, but just can't remember how I got there and it's been a while. Could someone help me…
Chris
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Derivative of CDF of inverse Gaussian distribution w.r.t. parameters

I am new user at this site need help in my research. I want to find the derivative of CDF of inverse gaussian distribution w.r.t. to parameters $\lambda$ and $\mu$. The PDF and CDF of inverse gaussian distribution is given as $f(x;\lambda, \mu )=…
Dev
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Distribution of weighted difference of two $\chi^2_n$ random variables

Let $X,Y \sim \chi^2_n$ be two independent $\chi^2$-distributed random variables with $n$ degrees of freedom. What is the CDF/PDF of $$ Z = aX - b Y $$ with $a>0, b>0$ ? P.S. This question answers the case $a=b=1$. My attempt: Using the moment…
PseudoRandom
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covariance of two linear combinations of a bivariate normal distribution

$X$ and $Y$ are jointly normal, with the mean vector and covariance matrix given by: $$\mu= \begin{pmatrix} 1 \\ 2 \\ \end{pmatrix} \Sigma= \begin{pmatrix} 2 & 0.4 \\ 0.4 & 1 \\ \end{pmatrix}$$ Let $Z_1=X+Y$ and $Z_2=2X-Y$. What is the mean vector…
woaini
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which families of PDFs are closed under translation and multiplication?

The product of two PDFs is not usually a PDF; but it is a (possibly zero) scalar multiple of another PDF. I will call a set $S$ of PDFs over $\mathbb{R}$ closed under translation and multiplication if $f(x)\in F \Rightarrow f(x-a) \in F$ for all…
user7530
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Sample from a bivariate lognorm

We suppose that X$\sim $(μ,Σ) with mean-vector μ=$(1.5,1.5)^t$, covariace matrix Σ=$\binom {1.5\quad 1.2}{1.2\quad 1.5}$ and $\boldsymbol{Y}=e^\boldsymbol{X}$. I would like to sample from a bivariate lognorm using R. By using the command…
Spy93
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Probability of Given Event and Cumulative Distribution Function

For the given Random Variable X, Cumulative Distributive Function is defined as below: $$F(x) = \begin{cases} 0, & \text{$x \le 0$} \\ x^2/8, &\text{$0\le x \lt 2$}\\ 1,& \text{$x \ge 2$} \end{cases} $$ And Let the two events $C_k, D_k$ be $$C_k =…
Beverlie
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Geometric Distribution: Probability of difficult or fair exams

Suppose a certain exam is classified as either difficult (with probability 90/92) or fair (with probability 2/92). Exams are taken one after the other. What is the probability that at least 4 difficult exams will occur before the first fair one?
AK47_93
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Poisson Distribution (Influenza)

The number $X$ of people in a certain town who get the influenza follows a Poisson distribution. The proportion of people who did not get the flu is $0.01$. Find the probability mass function of $X$. Ok so the probability mass function formula is…
AK47_93
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Understanding this probability distribution function

A certain country removed the $1$ cent and $2$ cent coin, and therefore was intended that every cash transaction be rounded to the nearest $5$ cent (note there are $100$ cents in every dollar.). Assuming that the number of cents in each transaction…
Natash1
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Finding distribution of a random variable using characteristic functions

Random variables $X_1,X_2,X_3$ are independent with distributions: $P\{X_1=k\}=1/2^k, P\{X_2=k\}=2/3^k, P\{X_3=k\}=4/5^k,k\in\mathbb N.$ Using characteristic functions, find distribution for random variable $X=X_1+X_2+X_3$. I will show the solution.…
user7920621
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distribution of sum of samples of a random variable

I have a random variable X (which has some complicated PDF, it could be approximated to exponential), and I'm trying to find the distribution of another random variable Y, which is the sum of j samples of (1/X) $Y_j=\sum_{i=1}^j(1/Xi)$ is there a…
A. Shokair
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Do all cumulative distributions have the property that it is equal to 1 outisde of the domain?

Relearning cumulative distribution functions and their definitions, it says that $F_x$ is a CDF if it is non-decreasing and right continuous with value of $1$ for large $x$ and value of $0$ for negatively large $x$. But this just says that it can…
Natash1
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