# Questions tagged [gamma-distribution]

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

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### Intuition behind relation between Gamma and Standard Normal distribution [migrated]

I read if $Z$ is a random variable with a standard Normal distribution and $X=Z^2$ then $X \sim Gamma(1/2, 1/2)$. I understand the math (manipulations of formulas) behind it. What about the intuition? ...
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### Integral involving exponential function and lower incomplete gamma function

Can we get the closed form value of the integral \begin{equation*} \int_{0}^{\infty}e^{-ax}\gamma(x,b)dx, \end{equation*} here, $a$ is a positive real numbers and $b$ is positive integer. Any ...
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### Derivation of Gamma distribution without using Poisson distribution

Most of the derivations of the Gamma distribution pdf I've seen on here use the Poisson distribution. My lecture notes use the Gamma distribution and the exponential inter-arrival time definition of a ...
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### Computing Posterior Distribution of Hyperparameter in a Multivariate Normal Model

I need guidance on computing the posterior distribution of a hyperparameter in a specific multivariate normal model. Here's a brief description of my problem: I have a dataset where the observed ...
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### Integration issue with the Gamma statistical model

I need to verify if an MLE is biased for this Gamma statistical model. \begin{align*} \mathbb{E}\left[\frac{1}{\bar{X}}\right]&=\int^\infty_0\frac{1}{s}\frac{(n\beta)^{2n}}{\Gamma(2n)}s^{(2n-1)}e^{...
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### Posterior distribution for gamma distributions

Im given the question: Given that the Gamma distribution likelihood $$p(y|β) = Gamma(y; α, β) = \frac{\beta^\alpha}{Γ(\alpha)} y^{α-1} e^{(-βy)}$$ where α is a positive constant and β > 0 is ...
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### Sum of Gamma distribution with different scale.

Let, $X_i$~$exp(\lambda_1)$ and $Y_i$~$exp(\lambda_2)$ iid for i = 1, 2, 3, .... Define the r.v, $Z_1^k = \sum_{i=1}^{k}(X_i + Y_i)$ and $Z_2^k = \sum_{i=1}^{k}(X_i + Y_i) + X_{k+1}$ for k=1, 2, 3, ....
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### Probability of exact k occurence within a fixed waiting time

Assume that the waiting time for each event follows the exponential distribution with the parameter $\lambda$. Let $\tau_{i,j}$ denote the waiting time between $i$th event and $j$th event. For a fixed ...
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### Gamma random variable multiplied with a negative constant

If X has a gamma distribution with shape parameter a and rate parameter b, then kX with also be gamma-distributed with rate parameter $\frac{b}{k}$. But this is only true if k is a positive constant. ...
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### Joint distribution of normal and inverse Gamma for full conditional distribution

Suppose $$x|\mu_1,\mu_2\sim N(\mu_1+\mu_2,1),$$ $$\mu_i\sim N(0,\sigma^2),\ iid$$ $$\sigma^2\sim Inv-Gamma(a,b).$$ and assume the first observed $x=1.$ We want to use Gibbs to sample $x.$ I already ...
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### Figuring out a probability distribution

At the instant $t = 0$ a certain radioactive focus starts emitting particles. The infinitesimal probability that the focus emits a particle in the differential interval is $\lambda dt$. Let $N$ also ...
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### Finding CLT for Gamma Distribution

It seems like a lot of examples for estimating the confidence intervals of a Gamma distribution, the parameter estimation involve one variable being known. I was wondering how to find a confidence ...
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### $X$ and $Y$ are independent with the same distribution and we know distribution of $X + Y$. Find distribution of $X$

We are given the following task: $X$ and $Y$ are independent random variables and have the same distribution. $U$ and $V$ are independent and $U, V \sim \Gamma(2,1)$. $X + Y$ has the same distribution ...
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### Estimate minimum number of spare parts

The life expectancy of a component of a machine is $20$ and its variance is $625$. If the component breaks it will be replaced instantanously by a new one. Estimate how many spare parts you need so ...
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### Nested Gamma Distributions

Let $a, c$ and $d$ be positive real numbers. Let us also assume that $b \sim \Gamma (c, d)$. Next, let us define random variable $X \sim \Gamma (a, b)$. Then we would like to know the probability ...
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### Why isn't $\frac{\beta^2}{\alpha^2}\int_{0}^p x^2 f(x, \alpha, \beta) dx = F(p, \alpha+2, \beta)$?
I'm look at the following integral: $$\int_{0}^p x^2 f(x,\alpha, \beta) dx$$ where $f(x, \alpha, \beta)$ is the pdf of the Gamma distribution is expressed as: \frac{x^{\alpha-1}e^{-\beta x}\beta^\...
Suppose we have an integral of this form: $\overline F(x)=\frac{\beta^{2\alpha-1}}{\Gamma^2(\alpha)}\int_{0}^{\infty}x^{\alpha-1}e^{-\beta(\frac{x}{y}+y)}dy$, where $\beta>0, \alpha>0$ and \$x>...