Questions tagged [gamma-distribution]

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

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Is the moment generating function of the gamma density $g(t)=(\frac{\lambda}{\lambda - t})^n$?

My book defines the gamma density as the following: $$f_X(x)=\lambda (\lambda x)^{n-1}e^{-\lambda x}/(n-1)!$$ And has the moment generating function of this density as $\frac{\lambda}{\lambda +t}$. Is ...
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1answer
48 views

Find probability distribution of $X+Y$

By knowing that \begin{equation} f_{(X,Y)}(x,y) = \begin{cases} \frac{1}{2} (x+y)e^{-(x+y)} & x,y>0 \\ 0 & \text{otherwise}\end{cases} \end{equation} I need to prove that $X+Y \sim \Gamma(3,...
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6 views

Approximation of $1-\left[\frac{\gamma(m_i,m_i\beta_i y)}{\Gamma(m_i)}\right]^{N_i}$.

I was working with gamma random variables. Let $Y_1$ the maximum of $N_1$ iid gamma random variables with parameter $m_1$ and $\beta_1$. Similar let $Y_2$ the maximum of $N_2$ iid gamma random ...
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2answers
36 views

How to derive the density of the square of a standard normal and chi-squared density from the gamma density?

My question is two parts. My book defines the gamma density as: $$g_n(x) = \lambda\frac{(\lambda x)^{n-1}}{(n-1)!}e^{-\lambda x},\space x>0$$ Part 1: In an example my book states that the ...
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0answers
10 views

Support of Tweedie Distribution as Function of P (Compound Poisson-Gamma)

My understanding of the Tweedie distribution is a bit limited. Let $X$ be a random variable with a Tweedie distribution with parameter $p$. If $p=1$, then $X$ is a Poisson distribution, which ...
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1answer
14 views

Variance of inverse gamma distribution

Given a random variable $X$ which is distributed gamma with shape $\alpha$ and rate $\lambda$, for which the variance is known, how does one calculate $\text{Var}(\frac{1}{X})$? I am hoping not to ...
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1answer
22 views

Finding negative 2nd moment of gamma distribution

Given a gamma distribution with shape $\alpha=2$ and rate $\lambda=10$, I was first asked to find an expression for $\Bbb E[X^k] \ \forall \ k \in \Bbb N$. Directly computing this, I got $$\Bbb E[X^...
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1answer
24 views

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
101 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
57 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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48 views

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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1answer
28 views

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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1answer
40 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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35 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
78 views

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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1answer
85 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
17 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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44 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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27 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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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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21 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
25 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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1answer
30 views

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
51 views

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
34 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
22 views

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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23 views

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
55 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
20 views

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
60 views

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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1answer
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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33 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
16 views

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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65 views

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
36 views

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
40 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
1k 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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0answers
36 views

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
77 views

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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43 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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1answer
53 views

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
122 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
28 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
35 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 ...