# Questions tagged [gamma-distribution]

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

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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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I want to prove that for standard normal vectors $x \in \mathbb{R}^n, x \sim \mathcal{N}(0, I_n), x' \in \mathbb{R}^{n + 1}, x' \sim \mathcal{N}(0, I_{n + 1})$: $$P\{\|x\|_2^2 > n \theta\} \... 1 vote 0 answers 26 views ### \mathbb E(e^{P(X)}) where X is gamma distributed and P a polynomial. Suppose that X is gamma distributed with scale s and shape \alpha, and that P is a polynomial of degree n :$$P(x)=\sum_{k=0}^n p_k x^k.$$When it exists (e.g., when the coefficients p_k ... • 276 0 votes 0 answers 19 views ### A problem involving a Normal approximation to a Gamma distribution I'm working on a course problem (unpublished), A marketing analyst is asked to investigate the distribution of user links within a social media platform. It is well known that a few users are very ... • 1,891 0 votes 0 answers 22 views ### Is Pareto(a,m) a subcase of Log-Gamma if m<1? I am approximating an empirical cumulative distribution (with domain [0,1]) with different curves e.g. Pareto and Log-Gamma. Since the distribution has domain [0,1] I need the Pareto (Par(\alpha,... 0 votes 0 answers 43 views ### Derivation of Characteristic function for Variance-Gamma process I'm trying to derive the characteristic function of a Variance-Gamma process, but I'm stuck. So, let X_{t} = \theta G_{t} + \sigma B_{G_{t}}, where B_{t} is a Brownian motion and G_{t} is a ... • 47 0 votes 1 answer 40 views ### Gamma Distribution Moments Derivation I got the following gamma function: \frac{B^{a}}{\Gamma(a)}x^{a-1}\ e^{-Bx}\ I_{[0,\infty)}(x) I would like to derive the raw moments and thus the variance but I know that I must have made a mistake ... • 59 1 vote 0 answers 133 views ### Nested Hierarchical Gamma-Poisson Model Introduction I have been studying the following hierarchical model$$ X_{i} \sim \begin{cases} \texttt{Po} ( \theta ), & i = 1, \dots , k; \\ \texttt{Po} ( \lambda ), & i = k+1, \dots , n, \... 79 views

### 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 ... 121 views

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### What happens to a Gamma distribution if it is shifted?

If $X \sim \operatorname{Gamma}(\alpha, \beta)$ then the transformation $kY$ also has a gamma distribution, i.e. $Y \sim \operatorname{Gamma}(k \alpha, \beta)$. Is there any known results to what ...
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### Gamma RV Transformations

$X_1, X_2$ are independent standard normals. Define $$W_1 = X_1^2 + X_2^2$$ and $$W_2 = (X_1^2 - X_2^2)/(X_1^2 + X_2^2)$$ How can I show that $W_1$ and $W_2$ are independent using the gamma ...
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### How to find the posterior distribution given data and prior distribution.

Suppose that following conditions are true: $$X\mid\Theta=\theta\sim N(0,\theta)$$ $$\Theta\sim InvGamma(2,1)$$ We need to find a posterior distribution: $$f\left(\Theta\mid X\right)$$ given the ...
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### Integration of density function [closed]

I am working with two random variables: $X\sim \text{Exp}(2\lambda)$ and $Y$, which has the following density function $$f_Y(x)=\frac{1}{\pi\sqrt{x(2-x)}},$$ where $0<x<2$. I'm trying to find ...
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### Prove a random variable to be a Chi-squared distribution

Here is the thing I am trying to prove: Suppose that the random variables $X_1, X_2, \dots, X_n$ are independent, and each random variable $X_i$ has a continuous c.d.f. $F_i$. Also, let the random ...
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### The summation of Gamma distribution

If ${X_1} \sim\gamma({\alpha _1},{\beta _1})$ and ${X_2} \sim\gamma({\alpha _2},{\beta _2})$, I need to prove ${X_1}+{X_2}∼Γ({\alpha _1}+{\alpha _1},{\beta _1}*{\beta _2})$ if ${X_1}$ and ${X_2}$ are ...
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### Proof: U, V follow the Gamma distribution with parameters p.

The question is : Let 0<q<p, suppose U follows the gamma distribution with parameters p, V follows the beta distribution with parameters q, p-q, and U and V are independent of each other. Proof: ...
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### Why does the unbiased statistic in this example be MVUE immediately?

I am reading "Introduction to Mathematical Statistics" to familiarize myself with sufficient statistics. I got stuck by an example in Sec. 7.3 of the book. Below are page 427 and 428 that ...
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I'm currently working on the following question: $$Define \: m=\sum_{i=1}^n log(\frac{\theta+X_i}{\theta}) \sim Pareto(n,\theta)\: and \: i.i.d$$ $$Let \: \hat{\alpha}=\frac{n}{\sum_{i=1}^n log(\frac{\... 0 votes 1 answer 28 views ### How do we derive the conditional distribution for a Poisson whose rate is the product of two Gamma distributed rv? This question is motivated by Gopalan et al. "Content-based recommendations with poisson factorization." Advances in neural information processing systems 27 (2014). https://proceedings.... 0 votes 0 answers 45 views ### With a Gamma(2, \frac{1}{2} ) use the CLT to prove the random variables  \sqrt{n}( \overline{X}_{n} - 1) \rightarrow_{d} N(0, \frac{1}{2} I'm currently noodling through a proof as to why a Gamma distribution of Gamma(2, \frac{1}{2} ) converges as per the Central Limit theorem:$$ \sqrt{n}( \overline{X}_{n} - \mu) \rightarrow_{d} N(0,...
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I'm dealing with two correlated random variables $d_1^2 = || r_1 - r_0 ||^2$ and $d_2^2 = ||r_2 - r_0||^2$, where $|| \cdot ||$ is the standard Euclidean distance ($L^2$ norm) and \$r_\cdot \sim \...