# Questions tagged [gamma-distribution]

For problems that are related to gamma-family probability distributions.

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### beta distribution as ratio gamma distributions

I need a proof of this statement please: Let $Y_1$ and $Y_2$ be independent random variables, where $Y_1$ is gamma distributed with parameters $\alpha$ and 1 and $Y_2$ is gamma distributed with ...
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### What is the probability of at least 3 goals in the last quarter of a 60 min game (given totally 5 goals)?

5 goals are scored in a 60-min hockey game (ignore breaks), and follow a Poisson process. What is the probability that at least 3 goals are scored in the last quarter of the game? I know when we have ...
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### Transforming from Gamma to Uniform distribution [duplicate]

Let X~ G($\alpha$,$\beta$) with the following pdf: f(x)=$\frac{\beta^\alpha}{\Gamma(\alpha)}$ $x^{\alpha-1}\cdot e^{-\beta x}$ I am supposed to find the pdf of new random variable Y~U[$0$,X]. How to ...
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### Why do waiting times in a queue tend to follow a gamma distribution?

I’m trying to get an intuition for gamma distributions, and why they are the model of choice for waiting times. In addition, I’d love to hear about any other distributions that are useful for modeling ...
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### Expected waiting time of last item for a set of m exponential random variables

So I've been mulling over a question: If I have a type of object that breaks after a waiting time $T \sim Exp(\lambda)$. Now I'm looking at $m$ of these same objects and I want to know the expected ...
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### Independence of the spacing of order statistics characterizes exponential distribution?

The question is: Let $Y_1 < Y_2$ be the order statistics of a random sample of size $2$ from a distribution of the continuous type which has p.d.f $f(x)>0$ provided $x \geq 0$, and $0$ elsewhere....
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### Asymptotic distribution of deviance residuals

While in a generalized linear model we do not assume Gaussian residuals, I seem to recall that the deviance residuals are normal asymptotically. Is this true (especially for the Gamma GLM) and does ...
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### Cumulative distribution for Poiss/Gamma conditional distributions

I have a problem compute the cumulative distribution $\pi(x,k)$ from conditional distributions. $$(K|X=x) \sim Poiss(x)$$ $$(X|K=k) \sim Gamma(k+1,2)$$ I implemented Gibbs sampling: ...
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### Use measured maximum and minimum to update gamma distribution

I found a report that states the number of samples $n$, mean $E(X)$, variance $V(X)$, and minimum $Min(X)$ and $Max(X)$ of a distribution. My prior belief is that this is a Gamma distribution. ...
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### Question about deciding the asymptotic distribution [closed]

Let $Y_n \sim \text{Gamma}(n, λ)$ where $n$ is an integer. As $n$ goes to infinity, find the asymptotic distribution of \begin{equation*} \sqrt{n}(\log(Y_n/n) - \log(1/\lambda)) \end{equation*} I ...
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### Literature on the uses and applications of Probability Distributions

I'm a graduate mathematics student and I've taken several courses on statistics, probability theory, stochastic processes and machine learning. In all the textbooks I consulted and all the classes I'...
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### Is there any significance of using the gamma function in the gamma distribution?

The gamma distribution models the probabilities of the time before $k$ Poisson events occur. So $k$ is restricted to integers here. Then why do we insist on using the gamma function instead of the ...
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### On Bayesian credible intervals

Question Let $X \mid \mu \sim \mathrm{Poisson} (\mu)$ and $\mu \sim \mathrm{Gamma} (1, 1)$ and suppose that a very large number $x$ is observed. Find, in terms of $x$, an approximate $95\%$ Bayesian ...
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### An expectation involving a Dirichlet distribution + the Gamma-representation seems to yield a strange integral expression/inconsistent results

This simple derivation involving the Dirichlet distribution is driving me crazy. Suppose that $\theta$ is a $m$-dimensional Dirichlet distributed vector with parameters $\alpha = \mathbf{1} \alpha_0$ ...
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### Compute the conditional probability distribution of a noncentral $\chi$ variable given the range of Erlang distributed non-centrality parameter

I need to compute a conditional probability distribution as described below for my research. In $(\mathbb R^2,||\cdot||_2)$, I have a random vector $\underline{z}$ with uniformly distributed angle and ...
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### Finding standarad error estimation/ Confidence interval of gamma parameter (say alpha) using central limit theorem

Suppose $\tilde \alpha$ is the MME of $\alpha$. Is there any way we can estimate the standard error or corresponding 95% CI of $\tilde\alpha$ using CLT. I have tried in this way [mean and variance are ...
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### Poisson Distribution vs Gamma distribution

Starting at 6 a.m., cars, buses, and motorcycles arrive at a highway toll booth according to independent Poisson processes. Cars arrive about once every 5 minutes. Buses arrive about once every 10 ...
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### Cannot find the constant $c$ if given $P(c\overline{X}<\theta)=0.95$.

Given $n$ random sample from exponential distribution which has probability density function \begin{align} f_X(x)= \begin{cases} \dfrac{1}{\theta}e^{-\frac{1}{\theta}x}& x>0,\theta>0\\ 0&...
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### The distribution of a sum of squared Gamma random variables

Random variable $X$ is a sum of $N$ squared Gamma RVs as: $$X=\sum_{i=1}^{N}X_i^2$$ where $X_i\sim Gam(k,\theta_i)$, the shape parameter $k$ is not an integer and scale parameters $\theta_i, \forall i$...
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### Gamma Distribution formulas

Why I am finding two formulas for the gamma distribution The one in my textbook is {1/(beta)^alpha*gamma(alpha)} * x^alpha-1 * exp(-x/beta) The one I found on the internet is {beta^alpha/gamma(...
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### Definite integral involving $\exp((-1/x) - x)$

I have a small integration problem. Just for context, this comes out of a bayesian exercise involving gamma functions. $$\int_{0}^{\infty} x^\alpha e^{-\left( \beta x^{-1} + \gamma x \right)} dx$$ ...
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### Gamma transformation

I am trying to find the distribution of $X/(X+cY)$ where X~$Gamma(n,\lambda/n)$ and Y~$Gamma(n,c\lambda/n)$ where c is a constant. I have applied the transformation w=X/(X+cY) and z=X+cY so that x=wz ...
Suppose I have two Gamma rvs $X1\sim G(m1,\Omega1)$ and $X2\sim G(m2,\Omega2)$ and both are i.i.d. Then how to derive the PDF of product of $X1\cdot X2$. Any help in this regard is highly appreciated.
May I ask a question regarding compound random variables, in which I want to obtain the distribution of $T$? $T=\sum_{i=1}^{N}t_i$, where $t_i$ are i.i.d exponential random variables with parameter \$\...