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Questions tagged [gamma-distribution]

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

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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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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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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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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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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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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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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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132 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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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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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 ...
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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{...
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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?
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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 ...
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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 ...
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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 $\...
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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 ...
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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 ...
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58 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'...
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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 ...
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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 ...
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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 ...
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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}{...
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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 ...
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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 $$F(t)= \frac{\Gamma_t(\alpha)}{\Gamma(\alpha)}, $$ ...
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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 ...
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31 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 ...
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73 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 $s=a(b+\frac{1}{4}+k)$. $k=0,1,2,...$ ...
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Making a statistical test for gamma distribution

I have a probability distribution based on an experiment results, which approximately follows gamma distribution with k=1 and θ=2 . Here I need to make a statistical test for testing on median(in this ...
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Conjugate Prior for Gamma Distribution

This is very basic, but I have been stuck on this problem for a while. Suppose $Y_1, \dots, Y_n|\alpha,\beta\sim Gamma(\alpha, \beta)$ is iid with $\alpha$ known. I want conjugate prior for $\beta$ ...
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Identifying a distribution by its properties

I am doing original research for my undergrad capstone course, and I was wondering if anyone here could help me see where to go next based on what I already know. I am trying to identify a ...
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115 views

Set the mode and median of a gamma distribution equal to each other

I am trying to generate a set of random positive steps that will result in a final location that is close to what I would have gotten from taking a similar number of fixed steps$^1$. I would like the ...
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Find E(X) for a certain function using the gamma function

The question I have to do is essentially this: A distribution, X, is modelled by $\displaystyle f(x)= \frac{x}{\sigma^2}e^{-x^2/2\sigma^2},\ x\ge0. $ Show that $\displaystyle E(X)=\sigma \sqrt{\...
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Is it possible to determine shape and scale for a gamma distribution from a mean and confidence interval?

Having the 95% confidence interval and mean for a distribution and knowing nothing else (other than the data is skewed and will likely follow a gamma distribution) is there any way to determine the ...
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Product of what iid random variables leads to a Gamma distribution?

I am trying to find out the distribution of $n$ i.i.d. random variables $X_1, ..., X_n$ so that their product $X_1 \cdot ... \cdot X_n$ follows a Gamma distribution. Or in other words, assume that $...
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Asymptotic of gamma function

I came across a quetion: Let $h$ go to zero. What is the asymptotic of $\Gamma(x+o_{p}(h))$ where $x\in(0,2)$? The difficulty is the limitation of x goes to zero. Can I obtain $$\Gamma(x+o_{p}(h))\...
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Trying to Understand $E[X^2]$ for Gamma Distribution

I am trying to understand the following for the gamma distribution: $$E[X^2] = \frac{ \alpha(\alpha+1)}{\lambda^2}$$ I've been looking at the reasoning for $E[X]$ to make sense of what could be ...
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How we arrive to the following form of CDF after SImlification

Let $N_1, N_2, m_1, m_2$ and $N_r$ positive integer numbers. Let $X_1$ and $X_2$ two independent random variables with CDFs \begin{align}\label{} F_{X_1}(x) =&\frac{1}{\Gamma(m_1)^{N_1}} \begin{...
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UMVUE of $\sqrt{a}/b$ for Gamma distribution

Suppose $(X_1,X_2,\ldots,X_n)\sim \operatorname{Gamma}(a,b)$, independent and identically distributed with pdf: $$f(x)=\frac{b^a}{\Gamma(a)}x^{a-1}e^{-bx},\quad x>0$$ Find the UMVUE of $\frac{\...
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For $Y\sim N(0,\sigma^2)$, find $\mathbb{E}(Y^n)$ for odd and even $n$ using the expectation of $G\sim \text{Gamma}(\alpha,\beta)$

For $\alpha,\beta>0$, the probability density function of a Gamma$(\alpha,\beta)$ random variable is given by $$f(x)=\frac{x^{\alpha-1}e^{\frac{-x}{\beta}}}{\Gamma(\alpha)\beta^\alpha} \ \ \ \ \ \...
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Finding the PDF of $\ T=\frac{\pi}{n}\sum_{i=1}^{n} X^2_i$

Consider a random samples $X_1,X_2,..,X_n$ from a variables with density function $$f_X(x)=2\lambda\pi xe^{-\lambda\pi x^2} \ \ \ \ \ \ \ x>0$$ I have shows that for $i=1,..,n$, $X^2_i\sim \...
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Show that for $i=1,..,n, \ X^2_i\sim \text{Gamma}(1,\frac{1}{\pi\lambda})$

Consider a random sample $X_1,X_2,..,X_n$ from a variable with density function $$f_X(x)=2\lambda\pi xe^{-\lambda\pi x^2}, \quad x\geq 0$$ I am trying to show that for $i=1,..,n,\ X^2_i\sim \text{...
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Show for $\ c>0\ $ that $\ cY\sim \ \text{Gamma}(\alpha,c\beta)$

Show for any constant $\ c>0\ $ that $\ cY\sim \ \text{Gamma}(\alpha,c\beta)$ $$Y\sim\text{Gamma}(\alpha,\beta)$$ $$f_Y(y)=\frac{1}{\Gamma (\alpha)\beta^\alpha}e^{\frac{-y}{\beta}}y^{\alpha-1} ...
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Calculating integral using gamma distribution

I've been studying form my Probability theory exam and I found this problem: Calculate using Central limit theorem $$\lim_{n\rightarrow\infty}\int_{0}^{n}\frac{1}{(n-1)!}x^{n-1}e^{-x}dx.$$ Using $$\...
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174 views

Gamma distribution shape parameter

Suppose that I have $X_i \sim E(1)$,iid, i goes from 1 to n, E stands for exponential distribution, and I want to know the distribution of $\bar{X} = \Sigma_iX_i/n$. I know that $\Sigma_i X_i \sim ...
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The PDF of a (Normal + Gamma-Gamma) Random Variable

In Free Space Optical (FSO) Communications systems, usually the turbulence is modeled as a random variable (r.v.) with Gamma-Gamma distribution. The thermal noise of the receiver is also a r.v. with ...
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How to derive the domain of Gamma function

I was reading about the Gamma function, however, I have some trouble to figure out why the domain of $$\Gamma(\alpha)= \int_0^{\infty} x^{\alpha -1}e^{-x}\,dx$$ How can I get $ \alpha > -1$?
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Inverse gamma distribution general question

I am reading a paper in the genomics field (Adjusting batch effects in microarray expression data using empirical Bayes methods. from W. Evan Johnson, Cheng Li), where they try to correct for some ...
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Gamma-distributed variable to the power $3$

If $X^\frac 13\sim\operatorname{GAM}(\theta,3)$ (so $\kappa=3$). Then what is the distribution of $X$? Is there any way to do this? I have tried by making use of the original MGF and doing this to the ...
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Difference of two gamma variables with same scale and shape

This question is similar to Difference of two Gamma random variables What is the probability density function of the difference of the gamma variables with the same scale and shape? The answer to ...