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Questions tagged [gamma-distribution]

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

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1answer
144 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
146 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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26 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 ...
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1answer
41 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{...
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2answers
46 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?
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1answer
61 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 ...
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22 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 ...
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0answers
79 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 $\...
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20 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 ...
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82 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 ...
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1answer
665 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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68 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 ...
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35 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 ...
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132 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 ...
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2answers
612 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}{...
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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 ...
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227 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 $$F(t)= \frac{\Gamma_t(\alpha)}{\Gamma(\alpha)}, $$ ...
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36 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 ...
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1answer
70 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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1answer
131 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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18 views

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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1answer
222 views

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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1answer
200 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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1answer
44 views

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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1answer
516 views

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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1answer
134 views

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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1answer
117 views

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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2answers
124 views

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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22 views

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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0answers
371 views

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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1answer
99 views

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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1answer
55 views

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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0answers
56 views

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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1answer
50 views

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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1answer
45 views

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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2answers
258 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 ...
2
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1answer
77 views

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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1answer
60 views

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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0answers
503 views

Beta distribution as ratio involving Gamma distributions

Let $X$ and $Y$ be two random variables such that $X \sim \text{Gamma}(a, \lambda)$ and $Y \sim \text{Gamma}(b, \lambda)$. Let $W$ be a new random variable such that $W := \frac{X}{X + Y}$. If $X$ and ...
3
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1answer
81 views

Exponentially Correlated Draws From Gamma Distribution

This is more about algorithms than math. I want to generate a series of random numbers corresponding to a given distribution, but in such a way that each draw is correlated to the previous according ...
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13 views

Indefinite Integral procedure

Can anyone explain me this definite integral I got from a probability problem: $\mathbf E[X] = 2\int_0^\infty x^2e^{-x^2} dx$ $[x^2 = z]$ $=\int_0^\infty z^{{3\over2}-1}e^{-z} dz$ $= \Gamma({3\...
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1answer
95 views

Finding the distribution of $xy$ when Distribution of $x$ and $y$ is given

If $x$ follows $Beta(u,v)$ and $y$ follows $Ganma(\lambda,u+v)$ then what is the distribution of $xy$? I could only write the PDF of x and Y but not able to draw PDF of XY. I suppose that there is ...
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0answers
34 views

X~Exponential(2) and Z~Exponential(2) added is Gamma(2,2)?

Okay, so I have this problem that states $X$~$Exponential(2)$, and $Z$~$Exponential(2)$ where Z is independent from $X$. The problem that I'm trying to solve wants me to find the distribution $X+Z$ ...
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0answers
763 views

Fisher information of reparametrized Gamma Distribution

I'm trying to solve the following problem: Let $X_1,...,X_n$ be iid from $\Gamma(\alpha,\beta)$ distribution with density $f(x)=\frac{1}{\Gamma(\alpha)\beta^\alpha}x^{\alpha-1}e^{-\frac{x}{\beta}}$....
1
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1answer
45 views

Does the convolution property of Gamma require independence?

I understand that when α is an integer, the Gamma(α,β) distribution is the distribution of the length of time you have to wait until a total of α events have occurred, where the time between each ...
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2answers
49 views

$\Gamma(\alpha+1,t)\geq (t+1)\Gamma(\alpha,t)$:Inequality related to incomplete gamma function

I am trying to prove an inequality which goes as For $\alpha\geq 1$,$\int_{t}^\infty u^{\alpha}e^{-u}du\geq (t+1) \int_{t}^\infty u^{\alpha-1}e^{-u}du$ for any $t\geq 0$. Using the notation of ...
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1answer
50 views

Conditional Probability Density Function of a single Random Variable With A Gamma Distribution

I am trying to find the probability density function of a Gamma distribution given that x > 4. I thought that I would be able to take the density and simply set it back 4 so that the domain would be ...