# Questions tagged [gamma-distribution]

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

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### Why does the unbiased statistic in this example be MVUE immediately?

I am reading "Introduction to Mathematical Statistics" edition 8 by Robert V. Hogg et al. to familiarize myself with sufficient statistics. I got stuck by an example in Sec. 7.3 of the book. ...
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### Alternating Renewal Process: How to calculate variance without knowing how the two distributions depend on each other

I am trying to solve a Alternating Renewal Process exercise. The "on" state follows a exponential distribution with mean 2. The time in the "off" state follows a gamma distribution ...
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### How to find p-value when null distribution is of gamma and given observation

Doing a hypothesis question, where $$H_0: \lambda=10$$ $$H_1: \lambda \neq 10$$ null distribution to be ~$\gamma(\alpha=20,\lambda=10)$, where 10 is in rate, and an observed sample =0.8, how do I ...
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### If $X\sim G(a,b_{1})$ and $Y\sim G(a,b_{2})$, then what will be the density function for U=min(X,X+Y)?

Let $X$ and $Y$ two independent random variables for gamma distributions with common shape parameter $a$ and different rate parameter $b_{1}$ and $b_{2}.$ If $U=\min(X,X+Y),$ then what will be the ...
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### Deriving a mixed distribution from exponential and inverse gamma

Question You are given the following: the amount of an individual loss in the year $2022$ follows an exponential distribution with mean $15000$ Between $2022$ and $2025$, losses will be multiplied by ...
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### Why does Gamma distribution appear in other distributions [closed]

Why does gamma distribution appear a lot? Gamma distribution appears in Chi-square distribution, Erlang distribution. It also appears in the expected value $(E[X])$ of Pareto distribution and Weibull ...
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### distribution of a transformed gamma random variable

I have $X=Gamma(a,b)$ and $Y=cX$ where c is a positive constant; I need to find the distribution of Y using the moment generating function method. I know $m_Y(t) = E e^{tY} = E e^{tcX}$ If i ...
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### The $q$'th moment of gamma distribution?

Let $X \sim \Gamma(\beta,\lambda)$ where $\beta>0$ is the rate paramter and $\lambda>0$ is the shape parameter. When I want to compute the $q$'th moment I get that \begin{align*} \mathbb{E}[X^q] ...
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### Sum of i.i.d. random variables

I am trying to obtain the distribution of $Y = \sum_{i=1}^{N} X_{i}$ when the distribution of each i.i.d. random variable $X_{i}$ is given by \begin{align} f_{X_{i}}(x) = \begin{cases}\lambda e^{\...
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### Baysian analysis of the Poisson distribution

$D = (x_i )_{i=1:n}$ is the training data, where $x_i$ follows a Poisson distribution of parameter $\lambda$. The likelihood is $p(D | \lambda) = \prod_{i=1}^n exp(-\lambda) \lambda^{x_i}/{x_i!}$ ...
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