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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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Can we simplify the integral $\int_0^\infty\cdots\int_0^\infty f\left(\sum_{i=1}^ns_i\right)g\left(\sum_{i=1}^n\alpha_is_i\right)ds_n\cdots ds_1$?

How can we simplify the integral $$I:=\int_0^\infty\cdots\int_0^\infty f\left(\sum_{i=1}^ns_i\right)g\left(\sum_{i=1}^n\alpha_is_i\right){\rm d}s_n\cdots{\rm d}s_1,$$ where $\alpha_1,\ldots,\alpha_n&...
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Derivative of incomplete gamma function w.r.t. the first argument

Wolfram Research provides the following formulas for the derivative of the incomplete gamma function: https://functions.wolfram.com/GammaBetaErf/Gamma2/20/01/01/0002/ https://functions.wolfram.com/...
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How can we compute the Expectation of Log of Truncated Gamma over (0, 1]?

How can we compute $\mathbb{E} \left[ \log \text{Gamma}_{(0, 1]} (\alpha, \beta) \right]$? Right now I'm using a Monte Carlo estimator by sampling from the Truncated Gamma distribution over $(0, 1]$ (...
ufer324's user avatar
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3 answers
76 views

Joint distribution of exponential random variable conditional on their sum

Let $Y_1, ..., Y_n$ be independently exponentially distributed with rate $\lambda$. My question: what is the joint distribution of $Y_1, ..., Y_n$ conditional on $T = \sum^n_{i=1}y_i$ ? We know that $...
Guillaume F.'s user avatar
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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 ...
Naveen Kumar's user avatar
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1 answer
38 views

Find unconditional distribution from conditional density functions of gamma and normal distributions

Let Y ∼ Gamma(a; λ) , a, λ > 0: that means random variable Y has continuous density: $$f_Y(y) = \begin{cases} \frac{\lambda^a y^{a-1} e^{-\lambda y}}{\Gamma(a)}, & y > 0, \\ 0, & \text{...
DavSilk's user avatar
2 votes
1 answer
64 views

When is the quantile function of a gamma distribution concave?

I am thinking about the consequences of adding prediction intervals and the consequence it has on the resulting interval. For example, I am considering when to expect the sum of two such intervals to ...
Galen's user avatar
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Deriving the Fisher information matrix for a reparameterised gamma distribution

Let $X \sim \mathrm{Gamma}(\alpha, \theta),$ where $$f(x) = \frac {x^{\alpha - 1} e^{-\frac x \theta}} {\theta^{\alpha}\Gamma(\alpha)}.$$ The log-likelihood function can be shown to be $$l(\alpha, \...
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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 ...
hegash's user avatar
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1 answer
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Predictive distribution for gamma likelihood and prior

For the following problem, I’m stuck at the 3rd question. I would appreciate if you could validate my answer to the 1st question as well. Problem Let $X_1, \dots, X_n \sim \operatorname{Gamma}(2, \...
Alexandre Huat's user avatar
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Finding the joint density function of $(X+Y,X)$ with $X$ and $Y$ independent and following an exponential distribution with parameter $\lambda>0$

Given two independent random variables $X$ and $Y$ that both follow an exponential distribution of parameter $\lambda > 0$, I am trying to find the joint density function of $(X+Y,X)$. I have ...
Gabriel Gontier's user avatar
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pdf of Linear Combination of the same random variable

Let's say that a random variable X has a probability p to be Gamma($\alpha,\beta$) and a 1-p probability to be $\chi^2$(r). How do I prove that $f_x(x) = p \frac{1}{\Gamma(\alpha)\beta^\alpha}x^{\...
Albert Wijaya's user avatar
6 votes
2 answers
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Insurance company with claims following a Poisson Process. Calculate the probability that the capital is always positive throughout the first 4 days.

Suppose that claims are made to an insurance company according to a Poisson process with rate $10$ per day. The amount of a claim is a random variable that has an exponential distribution with mean $1,...
Yash Jain's user avatar
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Modeling the probability of observing $N$ photons in an interval of time

Coherent beams produced by lasers follow Poissonian statistics, that is, the number of photons passing through the cross-section of the beam per unit time can be described by the Poisson distribution. ...
Aaron Hendrickson's user avatar
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1 answer
92 views

$X$ has gamma distribution with parameters $\alpha = \frac{1}{3}, \ \beta = n.$ Find $\lim_{n \longrightarrow \infty} P(|X_n-3n|<x\sqrt{n})$

Random variable $X$ has gamma distribution with parameters $\alpha = \frac{1}{3}, \ \beta = n,$ i.e. its pdf $p(y)=\frac{\frac{1}{3^n}}{\Gamma(n)}y^{n-1}e^{-\frac{y}{3}}$ when $y>0$, and $0$ ...
fragileradius's user avatar
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Asymptotic behavior of Regularized Gamma Function

I'm currently working on the regularized gamma function: $$ \frac{\gamma(x, f(x, c))}{\Gamma(x)} = c $$ Here, $c\in(0, 1)$ is a constant, and $f(x, c)$ is a function of $x$ and $c$ that satisfies the ...
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1 answer
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Magical relationship between Exponential distribution and Poisson process

Consider i.i.d. random variables $X_1,X_2,\ldots,X_n$ satisfying exponential distribution $\operatorname{Exp}(1)$. Let $Y=X_1+X_2+\ldots+X_n$. We know that the p.d.f. of $Y$ is the Gamma distribution $...
andy's user avatar
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1 answer
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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 ...
Dalek's user avatar
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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^{...
Jessie's user avatar
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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 ...
mads grønbeck's user avatar
-1 votes
1 answer
58 views

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, ....
ljh's user avatar
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2 votes
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Expectation of sde, need to calculate the probabilty of summation of gamma distribution.

Suppose $z(t)$ is the solution of sde. Let, $ \widetilde{z}(t) = z(t) - z(t_{2k})\textbf{1}_{k\neq 0}$ for $t_{2k}\leq t < t_{2k+1}$, $ \widetilde{z}(t) = 0$ for $t_{2k+1}\leq t < t_{2k+2}$ for $...
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Distribution of the division of two Gamma distributions.

Let $X \sim \Gamma(k,\theta)$ and Y $\sim \Gamma(p,\theta)$ What is the distribution of $Z = \dfrac{X}{X+Y}$ I know that there are some similar questions here on Stack, but these questions are related ...
Daileon108's user avatar
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66 views

The expectation of $\log(X)\cdot \mathbb{I}\{X\le a\}$, where $X$ follows gamma distribution

Given a random variable $X\sim {\text Gamma} (\alpha ,\beta )$, and define $Y=\ln(X)\cdot \mathbb{I}\{X\le a\}$, does it exist a closed-form expression for $\mathbb{E}\{Y\}$ or an approximated one? In ...
Lee White's user avatar
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113 views

Modifying Distribution of Sum of Independent Random Variables to Achieve a Specified Transformation

Consider a random variable $S$ defined as the sum of independent random variables: ${\displaystyle {S = X_{1} + X_{2} + ... + X_{n}}}$ ​As $n$ approaches infinity, the prob density of $S$ is given by $...
david's user avatar
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Distribution and expectation of Poisson random variable with gamma distributed mean

Question Suppose X ~ Pois($\lambda$), where the mean parameter $\lambda$ itself is a random variable with pdf $$g(\lambda;\alpha,\beta) = \frac{\beta ^\alpha}{\Gamma(\alpha)} \lambda^{\alpha - 1} \exp(...
Hmmmmm's user avatar
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3 votes
1 answer
151 views

Convex combination of Dirichlet random variables

For positive integer $k$, let $(X_1,\ldots,X_k)\sim\mathrm{Dir}(\alpha_1,\ldots,\alpha_k)$ be a probability distribution over $k$ items drawn from a $k$-component Dirichlet distribution and $p=(p_1,\...
user50394's user avatar
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Asymptotic behavior of Gamma cdf, gamma function, and incomplete gamma function

For the sake of this post, we only look at real-valued gamma functions. We know that $\lim_{b \to \infty}\gamma(a,b)=\Gamma(a)$, so we have $$\lim_{b \to \infty} \frac{\gamma(a,b)}{\Gamma(a)}=1.$$ How ...
Vergil's user avatar
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1 vote
1 answer
30 views

Numerical values of parameters of posterior distribution

Question Investigators are wishing to perform Bayesian analysis for drug treatment duration that can be described using a exponential distribution with paramater $\lambda$ describing drug duration in ...
DanielMariam's user avatar
1 vote
0 answers
231 views

Construction of confidence intervals for a Gamma random variable

Problem Suppose that $Y_1,Y_2,Y_3$ denote a random (independent) sample of size $3$ from a distribution with parameter $\lambda$ defined by the following probability density function: $$\begin{...
Hmmmmm's user avatar
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1 vote
1 answer
50 views

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 ...
Lupy's user avatar
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0 answers
92 views

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. ...
Thomas's user avatar
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1 vote
0 answers
105 views

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 ...
user6703592's user avatar
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1 answer
59 views

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 ...
user9867's user avatar
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0 answers
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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 ...
user215379's user avatar
1 vote
1 answer
293 views

$X$ 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 ...
Hinko Pih Pih's user avatar
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Does it converges to $0$?

I have to show that this expression zero $\frac{\log(\bar{F}(p^{-1}(x_n))}{n(1-x_n)^2} \to 0$ for $n\to \infty$, where F is the cdf of a gamma distribution with scale $k\cdot \alpha$ and rate $\frac{\...
kasper kaspersen's user avatar
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51 views

The distribution of 2 gamma random variables

If we have that both $X_1$ and $X_2$ are gamma(r,$\lambda$) distributed where $\lambda$ is the rate so the density is $\Gamma(r)^{-1}\lambda^rx^{r-1}e^{-\lambda x}$. What is the distribution of $-(...
kasper kaspersen's user avatar
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34 views

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 ...
Philipp's user avatar
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3 votes
1 answer
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For a standard normal vector $P\{\|x\|_2^2 > n \theta\}$ is monotonic in $n$

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\} \...
Mrs Robinson's user avatar
1 vote
0 answers
42 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 $p_k$ ...
lrnv's user avatar
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1 answer
330 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 ...
dewewdew's user avatar
1 vote
0 answers
168 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, \...
user avatar
2 votes
2 answers
140 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 ...
user avatar
2 votes
1 answer
131 views

Independence and Conditional Probability Density Function

Let us assume that $n \in \mathbb{N}$ and that $a_1 , a_2 , c_1 , c_2, d_1, d_2$ are positive real numbers. Let us define random variables $$ \theta \sim \Gamma (a_1, b_1 ) , \ \lambda \sim \Gamma ( ...
user avatar
1 vote
1 answer
118 views

Approximating $e^{-ax}-e^{-bx}$ to a gamma density function.

Is there an analytical approximation that results in the following: $\dfrac{x^{\theta-1}.(e^{-ax}-e^{-bx})}{\Gamma({\theta}).(a^{-\theta}-b^{-\theta})}\simeq \dfrac{x^{\theta^*-1}.e^{-cx}}{\Gamma({\...
Borna Bateni's user avatar
1 vote
1 answer
79 views

Summation of independent Gamma distributions?

Let $X_k$ be independent gamma distribution: $$X_k =\frac{X}{k}$$ where $X$ is a gamma distributions on $(0, \infty)$. How to find the distribution of the summation of $X_k$: $$\sum_{k=0}^{+\infty} ...
david's user avatar
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1 vote
1 answer
43 views

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^\...
Steven01123581321's user avatar
0 votes
2 answers
112 views

Asymptotics of tail function of product of 2 iid gamma variables

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>...
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