Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [gamma-distribution]

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

0
votes
1answer
26 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 ...
0
votes
1answer
37 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.
0
votes
0answers
10 views

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_{...
0
votes
1answer
17 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 ...
0
votes
2answers
19 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 ...
0
votes
0answers
17 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 ...
0
votes
1answer
29 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 ...
0
votes
1answer
10 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)...
0
votes
2answers
50 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 $\...
1
vote
1answer
63 views

A process converging to a $\Gamma$ distribution

What's an example of a discrete-time stochastic process that converges to a gamma distribution at equilibrium?
0
votes
0answers
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\...
0
votes
1answer
15 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 ...
0
votes
0answers
22 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 ...
0
votes
2answers
28 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}}...
0
votes
1answer
15 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 ...
0
votes
1answer
179 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 ...
1
vote
0answers
27 views

X follows gamma(1, $\alpha$ ) . What is the correct option [closed]

I dont know how the answers aren't dependent upon the value of alpha.
0
votes
0answers
12 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, $...
0
votes
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.
2
votes
1answer
54 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 ...
1
vote
1answer
22 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 $X$ ~ ...
0
votes
1answer
37 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.
0
votes
1answer
30 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 ...
1
vote
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 $$\...
3
votes
1answer
55 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 ...
0
votes
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 ...
0
votes
1answer
33 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 ...
0
votes
0answers
41 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 ...
2
votes
0answers
63 views

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(\...
5
votes
1answer
140 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(\...
5
votes
2answers
138 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 ...
1
vote
1answer
90 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 ) \...
0
votes
0answers
21 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 ...
1
vote
1answer
33 views

Convergence in probability: The inverse of the simple mean

I have a question on convergence: I have to prove that $\frac{n}{U_{n}} \longrightarrow 1$ in probability, where $U_{n}=\sum X_{i}$, $X_{i}\sim \mathrm{Exp}(1)$ and because of this, $U_{n}\sim \mathrm{...
0
votes
2answers
40 views

integrating gamma pdf over fixed limits

I am trying to solve $\int \limits _u^v x^{m-1}e^{-x} dx$. I checked table of integrals too but there is no direct solution for this, any help?
1
vote
1answer
47 views

Evaluating product of Upper Incomplete Gamma functions

I have checked several posts but couldn't find the equivalent of $\Gamma(m,a) \cdot \Gamma(m,b)$, where '$\cdot$' means multiplication. I suspect that it can be solved by applying the equivalent of ...
0
votes
0answers
17 views

What is the distribution of the integral of GBM on a finite support?

From this topic: Power of the integral of a Geometric Brownian motion I know that the random variable: $$ X = \int_0^\infty e^{aB_t-bt} dt $$ has the Inverse-Gamma distribution with some parameters (I ...
1
vote
0answers
53 views

Erlang Case of a Gamma Distribution

For part a) I get $ E(X)=\alpha\beta=\frac{n}{\lambda}. $ Thus the answer is $\frac{10}{0.5}=20$ minutes. I am not sure how to do b). Any help? The special case of the gamma distribution in which $\...
0
votes
0answers
18 views

Probability to create $n$ screens with a probability to have a breakdown

We need five successive working stations to produce a screen and the time spent on each of the working station is distributed as an exponential random variable. The average time of each working ...
0
votes
0answers
67 views

Supply chain modelling

So I have my first probability and statistics course this year, and we're currently learning about the different distributions that can be used to model, for example, supply chains. I was wondering ...
1
vote
1answer
290 views

Gamma Distribution Moments

Show that for X ~ Gamma($\alpha$, $\beta$), for positive constant $\nu$, $E[X^\nu] = \dfrac{\beta^\nu*\Gamma(\nu + \alpha)}{\Gamma(\alpha)}$. I have the following solution: Solution However, I don'...
2
votes
0answers
62 views

Average response/waiting time for aggregated tasks with Poisson arrival

Suppose there is a specific computation task with Poisson arrival rate $\lambda$ that could be aggregated in a way that when a task arrives and triggers a computation which lasts for $D$ seconds, if ...
0
votes
0answers
31 views

Two results obtained using Poisson distribution and gamma distribution approaches do not match up

This problem is exerpted from Walpole's probability book. The number of automobiles that arrive at a certain intersection per minute has a Poisson distribution with a mean of 5. Interest centers ...
0
votes
0answers
79 views

Find a pivot for a gamma distribution

I have some issues with an exercise where I have to find an approximate pivot for a gamma distribution... Indeed, I understand how to find a pivot for a normal distribution, but I don't understand how ...
0
votes
2answers
379 views

How did they get this proof for CDF of gamma distribution?

Let $$T \sim Gamma(\alpha, \lambda)$$ $$f(t) = \frac1{ \Gamma(\alpha)}{\lambda^\alpha}t^{\alpha-1}{e^{-\lambda t}} \qquad t,\alpha,\lambda > 0$$ The CDF result : $$F(t) = 1 - \sum_{i=0}^{\alpha-1}{...
0
votes
0answers
19 views

Why won't my Gamma difference density function run in R?

I'm trying to find the pdf of X where X is the difference of two iid Gamma distributions. the pdf is given in page 341 Theorem 2 of https://www.sciencedirect.com/science/article/pii/S0047259X83710365 ...
0
votes
0answers
147 views

How to find CDF of Gamma distribution for the Time (t) by integration?

I know the CDF of Gamma Distribution for the Time ($T \sim gamma(\alpha, \lambda)$) and shape $\alpha>0$ , rate $\lambda>0$ and $t>0$, is $$F(t)= \frac{\Gamma_t(\alpha)}{\Gamma(\alpha)}, $$ ...
0
votes
0answers
28 views

Distinguish between gamma and log-normal distributions based on 95th percentile of a random variable

I know mean and variance of a skewed positive random variable $X$ analytically. in literature both gamma and log-normal distributions can be fitted to such a random variable. I know that to find the ...
0
votes
1answer
40 views

Conditional probability with Gamma distribution.

I am stuck on a vital step in this process, and that is about the conditional probability aspect of it. Once I get that step, the integration and using the gamma function should go fine. So my ...
0
votes
1answer
96 views

inverse Laplace of Gamma function

what is inverse Laplace of following function? $$F(s)=L^{-1}(\frac{Γ(-\frac{s}{a}+b+\frac{1}{4})}{Γ(\frac{1}{4}-\frac{s}{a})})$$ I know this phrase has a pole in $s=a(b+\frac{1}{4}+k)$. $k=0,1,2,...$ ...