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Questions tagged [gamma-distribution]

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

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Compute the posterior on a Gamma distribution with a gaussian random variable

i have the following problem: given a variable $x \in \mathbb{R}$ drawn from a Gaussian distribution with known mean $\mu$ and unknown precision $\tau$ (the inverse of the variance). So: $$p(x \...
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3answers
67 views

If $Y\sim\operatorname{Beta}(a,1-a)$ and $Z\sim\operatorname{Exp}(1)$, then $YZ\sim\operatorname{Gamma}(0,1)$?

I have two random variables $Y \sim \operatorname{Beta}(a, 1 - a)$ $Z \sim \operatorname{Exp}(1)$ If $Y$ and $Z$ are independent, why is the distribution of $X = YZ \sim \operatorname{Gamma}(...
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1answer
41 views

Transforming sum of n exponential distribution to a Poisson distribution

Let $X_1,...,X_n$ be i.i.d exponential random variable with mean $\lambda$ $S=X_1+...+X_n$ So by finding the mgf of S, we get that $S \sim \operatorname{Gamma}(n,\lambda)$ The problem I am stuck ...
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How to prove the inequality? [closed]

Prove $\int_{n+2 t\sqrt{n}}^{+\infty} \frac{\left(2 t^{2}+x\right)^{\frac{n}{2}-1} e^{-\frac{x}{2}}}{2^{\frac{n}{2}} \Gamma\left(\frac{n}{2}\right)} d x \leq 1$ When $ t \geq 0 $ and $n \in \mathbb{N^+...
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Prove for $|t|<\lambda$ that for $\alpha > 0$ that $\varphi_{\Gamma(\alpha,\lambda)}(t) = \bigg(\frac{\lambda}{\lambda - it}\bigg)^{\alpha}$

From convolutions it's easy to see that for integer $n>0$ \begin{equation} \varphi_{\Gamma(n,\lambda)}(t) = \bigg(\frac{\lambda}{\lambda - it}\bigg)^n \end{equation} In fact for any real $\alpha &...
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1answer
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Let X be$Gamma(\alpha, \lambda)$Prove $(\lambda X - \alpha)/\sqrt{\alpha} \xrightarrow{d} N(0,1)$ as $\alpha \rightarrow \infty$ and $\lambda$is fixed [closed]

First of all the continuity lemma is stated as follows: Let $\mu_n, n=1,2, \dots$ be a sequence of distributions, and $\varphi$ the associated characteristic function. If $\mu_n \xrightarrow{w} \mu$,...
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38 views

Integration of gamma function

Insurance company has to pay payments at the rate of $d$ per year. They are payable continuously as long as the person remains sick. The length of the payment period in years is a random variable ...
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1answer
28 views

Gamma Distribution and Chi Square Distribution

Actally i have 2 question for you guys. It's not a homework. I just curious. What is the difference of Erlang distribution and Gamma distribution. On Wikipedia it's said if Erlang is Gamma ...
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3answers
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How do I evaluate the following combination of random variables? Is it martingale?

I'm about to analyse the following expression $$Z_n:=\prod_{k=1}^n \left(\frac{\frac{Y_k}{\prod_{i=1}^k X_i}}{\sum_{j=1}^k \frac{Y_j}{\prod_{i=1}^j X_i}} \right),$$ where $Y_j$ for all $j\in \mathbb{...
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41 views

A relationship between Poisson distribution and gamma distribution

We define $N(t)$ to be number of events in the interval $[0,t]$. We assume that $N(t) \sim P(\lambda t)$ for $\lambda > 0$. Let $X$ be the waiting time until the $n$-th event, we need to prove that ...
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0answers
16 views

Sufficiency in the exponential distribution

I am trying to show that given a random sample $\{X_i\}_{i=1}^n$ where $X_i\sim exp(\lambda^{-1})$, the statistic $T(\mathbf{X})=\sum_{i=1}^n X_i$ is sufficient by using only the definition. I have ...
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16 views

Introduction of shape parameters in the formulation of probability distribution

I'm familiar with the definition of location, scale, and shape parameters, and the type of distributions they parametrized. I'm interested in understanding how shape parameters became part of the ...
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43 views

Understanding The posterior distribution for a given model if it has some prior?

I was studying the posterior distribution and came across a question and didn't understand. What is the posterior distribution for a given that if a model has the following prior, $$𝑥_1, 𝑥_2,\dots,...
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26 views

How To Find the Probability of a Gamma Distributed Random Variable?

Suppose a random variable has a gamma distribution with $\alpha = 0.8$ and $\beta = 2.4$ How can we calculate $P(Y > 3)$? My book says when $d$ and $c$ are such that $0 < c < d < \infty$...
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1answer
44 views

Prove that $\sum^n_{i=1} X_i$ and $\prod^n_{i=1} X_i$ are sufficient statistics for the gamma distribution

This question is set in the statistical context, but my difficulty is more ‘pure math’ in nature, so I have posted it here instead of at the statistics forum. I am to prove that $V := \sum^n_{i=1} ...
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0answers
23 views

The Exact Confidence Interval for an MLE of a Gamma Distribution

Above here is the information I've been given for one of my seminar questions, so far I have calculated the fisher information and from there I computed the asymptotic distribution for $\hat{\lambda}$ ...
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0answers
16 views

EM algorithm for a Gamma mixture model

I am trying to understand and write an EM algoithm to estimate the components of a Gamma mixture model. I would like to know, at the M step, how do I update the parameters (shape, rate/scale) for ...
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0answers
23 views

CLT and sum of gamma random variables

I am having trouble approximating the sum of gamma-distributed variables via CLR. I know via Gamma that $X=\sum_{i=1}^n X_i \\$ and $X\sim\Gamma(n\alpha,\beta) \\$ and $CLT: Z_{n}=\frac{\overline{X}-...
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Bayes rule on conditional probabilities that follow Poisson/Gamma

Posting a problem i have to solve, just trying to understand how bayes works on conditional probabiliries (a) Assume X follows a Poisson distribution P(X = x|λ) = e^−λ * λ^x / x! , where the ...
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2answers
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How to calculate the limit $\lim_{n\rightarrow \infty} \sqrt 2 \frac {\Gamma\left(\frac {n+1} 2\right)} {\Gamma\left(\frac {n} 2\right)}-\sqrt n$

$$\lim_{n\rightarrow \infty} \sqrt 2 \frac {\Gamma\left(\frac {n+1} 2\right)} {\Gamma\left(\frac {n} 2\right)}-\sqrt n$$ It seems that using Stirling approximation doesn't work.
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The internally studentised residuals $r_i$

Suppose $Y = X \beta + \epsilon$ where $\epsilon \sim N(0, \sigma^2I)$. Show that for the internally studentised residuals $r_i$ defined as $$ r_i = \frac{\hat{\epsilon}_i}{\hat{\sigma}\sqrt{1-h_{...
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1answer
24 views

conditional expectation of gamma distribution with alpha = 1 [closed]

$X_i$ are exponential(\lamda) distribution and identically independent distribution. Y = $\sum_{i=1}^n$$X_i$ $X_i$ is an unbaised estimator of \lamda. Y is a sufficient estiamtor of \lamda. solve ...
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2answers
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Computing the derivative of the CDF of a gamma random variable

In class, my professor computed the density of a gamma random variable by taking the derivative of its CDF, but he skipped many steps. I am trying to go through the derivation carefully but cannot ...
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What is the random variable for this Gamma Distribution problem?

I am trying to solve this problem, and I know that this problem is solved using gamma distribution, but I am not sure how it works... A bakery sells rolls in units of a dozen. The demand $X$ (in ...
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1answer
40 views

Queuing processing and Gamma distribution

I've been trying to solve the following exercise and I was hoping for your input. If $Q$ is a queueing process with arrival rate $\lambda$ and service rate $\mu$, and a customer arrives to find ...
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1answer
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How to scale a generalized gamma distribution?

Is the following derivation for scaling a Generalized Gamma distribution correct? Given $X\sim GG(x;a,d,p)$, with $x\ge 0$, $a,d,p > 0$ and pdf $$f_x(x;a,d,p) = \frac{px^{d-1}exp\Big(-(x/a)^p\Big)...
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2answers
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Expected value of a Gamma RV to the power of a Poisson RV

$\mathit{W}$ is a $\bigl(\alpha = 3, \beta = \frac 12 \bigr)$ -Gamma random variable, and $\mathit{N}$ is a $\mu$ = $\frac 13$ -Poisson random variable, independent from $\mathit{W}$. What is $\...
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72 views

A process converging to a certain distribution [closed]

How is it possible to build a discrete-time stochastic process so that converges to a specific distribution at equilibrium, for example an exponential distribution or a gamma distribution ?
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32 views

Closed form for an integral involving a generalized incomplete Gamma function?

I am trying to find a closed form for this integral: $$\int_{0}^{\infty}\int_{0}^{\infty}e^{-d_{p,s}^v\,x-d_{s,p}^v\, y+d_{p,p}^v\,\frac{\xi_1\,\sigma^2\,xy}{P_p\,\xi_2\,xy+\sigma^2}}\mathrm dy\...
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1answer
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Marginal distribution of $X$ when $X|m \sim Pois(m)$ and $M \sim \Gamma (2,1) $

I am trying to find the marginal distribution of the joint $X$ and $M$ in order to find the probability $$Pr[X = 0,1,2,3]$$ I am given that $X|m \sim Pois(m)$ and $M \sim \Gamma (2,1) $ so I am ...
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Proof of relation of Gamma and Beta distribution without using Jacobian

If X1 and X2 independent random variables follow Gamma Distribution, can we prove Y= X1/(X1+X2) is a Beta Distribution without using the Jacobian Change of Variable method? In our course, we haven't ...
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2answers
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Integral involving Gamma distribution

I need some help with an integral. This is the solution to one of the problems I had to do. Everything is fine, but I don't understand one step: Now how is $$\int_0^\infty \frac{\beta_n^{\alpha_n+k}}...
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1answer
26 views

estimate Gamma parameters based on mean and variance

I am following these two approaches (which are the same)this and this, to estimate the two parameters of Gamma dist based on mean and var. I am not sure why I cannot get the same mean and var from ...
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1answer
866 views

Method of moments with a Gamma distribution

I'm more so confused on a specific step in obtaining the MOM than completely obtaining the MOM: Given a random sample of $ Y_1 , Y_2,..., Y_i$ ~ $ Gamma (\alpha , \beta)$ find the MOM So I found the ...
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Normalizing a Modified Gamma Distribution with limits

I am using a modified gamma distribution (MGD) for a particle size distribution of the form $$n(D) = N_0 D^\mu e^{-\Lambda D^\gamma}$$ where $N_0$ is the scale factor, $D$ is the particle diameter, $...
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1answer
44 views

Deduced distribution of X from X^2

If I have a random variable $X^{2}$ with distribution $X^{2}∼Γ(α,β)$ then what would be the distribution of $Y=X$ ? Thank you.
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1answer
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Understanding the connection between the chi-square and the gamma distribution

If $Z_1,\ldots, Z_n$ are independent standard normal random variables, then the random variable $X = \sum_i Z_i^2$ is said to have a chi-square distribution with $n$ degrees of freedom. If you ...
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1answer
23 views

Let $f(x,y) = \frac{1}{n!}(x-y)^n e^{-x} (0<y<x)$ be a joint density function

Let be $f(x,y) = \frac{1}{n!}(x-y)^n e^{-x} (0<y<x)$ a joint density function. I want to find the joint density function of $(U,V)$ where U = X and V = $e^{X-Y}$. Now, I already found that $X$ ~ ...
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1answer
40 views

If $\ln(x)$ is gamma distributed, what is the distribution of $x$?

Additionally, if someone could help calculate the mean and variance of $X$, that would be greatly appreciated.
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Searching for proof - bayesian inference for exponential distribution

According to Wikipedia (https://en.wikipedia.org/wiki/Conjugate_prior) the gamma distribution is a conjugate prior for the exponential distribution (with unknown rate-parameter, $\lambda$, and ...
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1answer
51 views

Numerical solution to a system of equations

Let $n\in\mathbb{N}$ and $u_1,u_2,\ldots ,u_n,t_1,t_2\geq 0$ be constants. I'm interested in finding the numerical solution in relation to $\alpha$ and $\beta$ to the following system of equations $$\...
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1answer
109 views

If $X_n$ is Gamma $(n,\lambda)$ distributed then $(\lambda X_n -n)/\sqrt n\to N(0,1)$

Let $X_n$ be Gamma $(n,\lambda)$ distributed, and $Y_n = \dfrac{\lambda X_n -n}{\sqrt{n}}$. Show that $Y_n \rightarrow N(0,1)$. My idea to prove this is to use Lévys theorem with the ...
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1answer
25 views

Find the PDF of gamma distributed random variable using derivation

Let $X$ be a random variable with CDF $F_X(x)$ given by $$ F_X(x)=1-\frac{\Gamma(m,(m/y)x)}{\Gamma(m)}, $$ where $m$ and $y$ are positive integers $(m>0, y>0)$ and $\Gamma(a,z)$ is the ...
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1answer
34 views

Using change of variables to transform density functions

I'm was working on some exercises on statistical inference and came across a question I could not solve. After a while I decided to take a look at the solution to hopefully understand the problem ...
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45 views

How can I solve this integral equation with the inverse Laplace Transform?

This question is related to Solving an integral equation with inverse Laplace transform. Let $\alpha,\beta,\mu>0$ with $\alpha/\beta>\mu$ and $X\sim\operatorname{Gamma}(\alpha,\beta)$. I am ...
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Characteristic Function of Gamma Distributed Random Variables

I have the following characteristic function $$\sum_{m=0}^{\infty} \frac{(is)^m}{m!} \sigma_{m,k} \frac{\Gamma(\beta + m)}{\Gamma(\beta)},$$ where $i$ is the imaginary unit, $\beta>0$, $\Gamma(\...
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1answer
143 views

Solving an integral equation with inverse Laplace transform

Let $\alpha,\beta,\mu>0$. I am looking for a solution, i.e. a function $g(x)$, that satisfies $$ \frac{\beta^{\alpha}}{\Gamma(\alpha)}\int_0^\infty g(x)x^{\alpha-1}e^{-\beta x}\,\mathrm dx=\left(\...
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2answers
139 views

Seeking Methods to solve $F\left(\alpha\right) = \int_{0}^{1} x^\alpha \arcsin(x)\:dx$

I'm looking for different methods to solve the following integral. $$ F\left(\alpha\right) = \int_{0}^{1} x^\alpha \arcsin(x)\:dx$$ For $\alpha > 0$ Here the method I took was to employ ...
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1answer
137 views

Central limit theorem for sequence of Gamma-distributed random variables.

Suppose that $X_ n \sim \text {Gamma}\ (n\alpha , \lambda)$ for all $n \ge 1$, for fixed $\alpha,\lambda >0.$ Show that $$\frac {1} {\sqrt n} \left (X_n - \frac {n \alpha} {\lambda} \right ) \...
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0answers
25 views

hypothesis testing - gamma distribution

Let W = Y/B0 be a Random variable that has a gamma(2n,1) distribution. [Y has a gamma(2n,B) distribution and W = Y/B]. i) Suppose you want to test H0 : B ≤ B0 against H1 : B > B0 for some B0 > 0. How ...