Questions tagged [gamma-distribution]

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

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18 views

Scalar multiple of a gamma random variable

Let $X \sim \mathrm{Gamma}(\alpha, \beta)$, where $\alpha$ and $\beta$ are the shape and rate parameters respectively. If $X_i \stackrel{\mathrm{iid}}{\sim} \mathrm{Gamma} (\alpha_i, \beta), \mathrm{...
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21 views

Inverse Gamma Distribution with Newton's method

I want to generate Gamma random variables using the inverse transform method. For this purpose I want to derive the inverse of the CDF of Gamma using the Newton's method. This method may be not so ...
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16 views

Simulating Gamma distribution MATLAB

I am a beginner in Matlab and I need an explanation for this. I am trying to generate random variables that are gamma distributed and compare them with the output of gamcdf. ...
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23 views

What Variance equation is this textbook using?

I'm doing a question on a Gamma/Poisson mix, which turns into a negative binomial. What I don't understand is the answer to the question - most of the answer makes sense to me (normal approximation ...
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18 views

Finding power-series expression for the probability density function given only expressions for its moments (which involve non-elementary functions)

In the field of radar, Swerling provided a relatively simple model to determine the probability of detecting an object given certain parameters about the object and your radar (call these parameters $...
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beta distribution as ratio gamma distributions

I need a proof of this statement please: Let $Y_1$ and $Y_2$ be independent random variables, where $Y_1$ is gamma distributed with parameters $\alpha$ and 1 and $Y_2$ is gamma distributed with ...
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31 views

What is the probability of at least 3 goals in the last quarter of a 60 min game (given totally 5 goals)?

5 goals are scored in a 60-min hockey game (ignore breaks), and follow a Poisson process. What is the probability that at least 3 goals are scored in the last quarter of the game? I know when we have ...
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15 views

Transforming from Gamma to Uniform distribution [duplicate]

Let X~ G($\alpha$,$\beta$) with the following pdf: f(x)=$\frac{\beta^\alpha}{\Gamma(\alpha)}$ $x^{\alpha-1}\cdot e^{-\beta x}$ I am supposed to find the pdf of new random variable Y~U[$0$,X]. How to ...
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45 views

Why do waiting times in a queue tend to follow a gamma distribution?

I’m trying to get an intuition for gamma distributions, and why they are the model of choice for waiting times. In addition, I’d love to hear about any other distributions that are useful for modeling ...
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1answer
23 views

Expected waiting time of last item for a set of m exponential random variables

So I've been mulling over a question: If I have a type of object that breaks after a waiting time $T \sim Exp(\lambda)$. Now I'm looking at $m$ of these same objects and I want to know the expected ...
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1answer
25 views

Independence of the spacing of order statistics characterizes exponential distribution?

The question is: Let $Y_1 < Y_2$ be the order statistics of a random sample of size $2$ from a distribution of the continuous type which has p.d.f $f(x)>0$ provided $x \geq 0$, and $0$ elsewhere....
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Asymptotic distribution of deviance residuals

While in a generalized linear model we do not assume Gaussian residuals, I seem to recall that the deviance residuals are normal asymptotically. Is this true (especially for the Gamma GLM) and does ...
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5 views

Cumulative distribution for Poiss/Gamma conditional distributions

I have a problem compute the cumulative distribution $\pi(x,k)$ from conditional distributions. $$ (K|X=x) \sim Poiss(x) $$ $$ (X|K=k) \sim Gamma(k+1,2) $$ I implemented Gibbs sampling: ...
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Use measured maximum and minimum to update gamma distribution

I found a report that states the number of samples $n$, mean $E(X)$, variance $V(X)$, and minimum $Min(X)$ and $Max(X)$ of a distribution. My prior belief is that this is a Gamma distribution. ...
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58 views

Question about deciding the asymptotic distribution [closed]

Let $Y_n \sim \text{Gamma}(n, λ)$ where $n$ is an integer. As $n$ goes to infinity, find the asymptotic distribution of \begin{equation*} \sqrt{n}(\log(Y_n/n) - \log(1/\lambda)) \end{equation*} I ...
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1answer
66 views

Literature on the uses and applications of Probability Distributions

I'm a graduate mathematics student and I've taken several courses on statistics, probability theory, stochastic processes and machine learning. In all the textbooks I consulted and all the classes I'...
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Is there any significance of using the gamma function in the gamma distribution?

The gamma distribution models the probabilities of the time before $k$ Poisson events occur. So $k$ is restricted to integers here. Then why do we insist on using the gamma function instead of the ...
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On Bayesian credible intervals

Question Let $X \mid \mu \sim \mathrm{Poisson} (\mu)$ and $\mu \sim \mathrm{Gamma} (1, 1)$ and suppose that a very large number $x$ is observed. Find, in terms of $x$, an approximate $95\%$ Bayesian ...
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Logarithm of Gamma(sum(alpha_i))

I have $\mathbb{Y}=(Y_1,Y_2, \dots Y_k)^T$ and need to compute $E(ln(Y_1))=D log(\frac{\prod_{i=1}^k{\Gamma(\alpha_i)}}{\Gamma(\sum_{i=1}^k \alpha_i)})$, $\mathbb{\alpha}=\alpha_1,\alpha_2,\dots, \...
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X follows the Gamma distribution with $\alpha=5$, calculate $\beta$ if $E(X^2)=11.7187500$

The entire question is in the title, however, I am posting this because I solved it and something seems off, hence the "solution-verification" tag this is posted under. My solution: We know ...
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32 views

Four students are giving presentations [duplicate]

In four sections of a course, running (independently) in parallel, there are four students giving presentations that are each Exponential in length, with expected value of 10 minutes each. How much ...
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1answer
61 views

Is it possible to use MGFs for find the distribution of $X/(X+Y)$ when $X$ and $Y$ are independent and gamma distributed?

Suppose that $$Z=\dfrac{X}{X+Y}$$ $$X \sim Gamma(a,\lambda)$$ $$ Y \sim Gamma(b,\lambda)$$ with $X$ and $Y$ independent. I would like to see if it might be possible to determine the distribution of $...
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An attempt to intuitively derive the Gamma distribution.

I study a course in probability theory these days and i have came across the notion of the Gamma distribution, which describes the probability distribution of the sum $Z_n=X_1+X_2+\cdots+X_n$, where $...
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Let $Y$~$Gamma(a,b)$ with PDF $f(y)$= $y^{a-1}e^{-y/b}/\gamma(a)b^a$; $y>0$ For $a>1$, show that the mode of $Y$ is $(a-1)b$.

Suppose that $Y$ follows a gamma distribution with parameters $a$ and $b$. That is, $f(y)$= $y^{a-1}e^{-y/b}/\gamma(a)b^a$; $y>0$ For $a>1$, show that the mode of $Y$ is $(a-1)b$. My working: I ...
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1answer
40 views

Product of two densities

Let $X \sim \Gamma(\alpha, \lambda)$ and $Y \sim \Gamma(\beta, \lambda)$. I denoty by $f_X$ the density of X and by $f_Y$ the density of Y. Additionally, I assume that the density of (X, Y) is $f_X \...
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25 views

An expectation involving a Dirichlet distribution + the Gamma-representation seems to yield a strange integral expression/inconsistent results

This simple derivation involving the Dirichlet distribution is driving me crazy. Suppose that $\theta$ is a $m$-dimensional Dirichlet distributed vector with parameters $\alpha = \mathbf{1} \alpha_0$ ...
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28 views

understanding the convolution of random variable formula

Consider summing two iid exponential r.v. We know for a fact that this is Gamma distribution with $\alpha = 2, \beta = \theta$. However, when using the convolution formula $Z=X+Y$, we have $$ f(z)= \...
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2answers
134 views

Finding $\mathbb{E}[X^s e^{tX}]$ where $X \sim \mathrm{Gamma} (\alpha, \beta)$

Question Find $\mathbb{E}[X^s e^{tX}]$, where $X \sim \mathrm{Gamma} (\alpha, \beta)$ and $\alpha$, $\beta$ are the shape and scale parameters respectively. Specify also the real values of $s$ and $t$ ...
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1answer
34 views

Covariance between exponential and gamma distribution

Define $W_{r}=\min\{t:N_{t}\geq r\}(r=1,2,...)$ as the waiting time until the $r$th arrival in a Poisson process $\{N_{t}:t \geq 0\}$ with rate $\lambda$. The question is to find $Cov(W_{1},W_{r})(r\...
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20 views

What is the distribution for a random variable compose with another random variable?

I have a a random variable $X$ with distribution $Po(M)$ where $M$ is a random variable with distribution $Exp(a)$. I want to know what is the "real" distribution of $X$. I was calculating ...
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2answers
39 views

Obtain $P\left( \sum_{i=1}^{k} Y_{i}<X<\sum_{i=1}^{k+1} Y_{i}\right)$ for $X$ and $Y_{i}$ independent exponential RVS

Given that $X$ is an exponential random variable with parameter $\lambda$ and $ Y_{i}$ are independent and identically distributed exponential random variables with parameter $\beta$. Given that X and ...
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1answer
77 views

What is the reasoning used here to determine the UMVUE?

Let $X_1, \dots, X_n$ denote a random sample from the PDF $$f_{\varphi}(x)= \begin{cases} \varphi x^{\varphi - 1} &\text{if}\, 0 < x < 1, \varphi > 0\\ 0 &\text{otherwise} \end{...
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60 views

Compute the conditional probability distribution of a noncentral $\chi$ variable given the range of Erlang distributed non-centrality parameter

I need to compute a conditional probability distribution as described below for my research. In $(\mathbb R^2,||\cdot||_2)$, I have a random vector $\underline{z}$ with uniformly distributed angle and ...
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1answer
29 views

Finding standarad error estimation/ Confidence interval of gamma parameter (say alpha) using central limit theorem

Suppose $\tilde \alpha$ is the MME of $\alpha$. Is there any way we can estimate the standard error or corresponding 95% CI of $\tilde\alpha$ using CLT. I have tried in this way [mean and variance are ...
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2answers
54 views

Poisson Distribution vs Gamma distribution

Starting at 6 a.m., cars, buses, and motorcycles arrive at a highway toll booth according to independent Poisson processes. Cars arrive about once every 5 minutes. Buses arrive about once every 10 ...
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Cannot find the constant $c$ if given $P(c\overline{X}<\theta)=0.95$.

Given $n$ random sample from exponential distribution which has probability density function \begin{align} f_X(x)= \begin{cases} \dfrac{1}{\theta}e^{-\frac{1}{\theta}x}& x>0,\theta>0\\ 0&...
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1answer
60 views

The distribution of a sum of squared Gamma random variables

Random variable $X$ is a sum of $N$ squared Gamma RVs as: $$X=\sum_{i=1}^{N}X_i^2$$ where $X_i\sim Gam(k,\theta_i)$, the shape parameter $k$ is not an integer and scale parameters $\theta_i, \forall i$...
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1answer
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Gamma Distribution formulas

Why I am finding two formulas for the gamma distribution The one in my textbook is {1/(beta)^alpha*gamma(alpha)} * x^alpha-1 * exp(-x/beta) The one I found on the internet is {beta^alpha/gamma(...
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1answer
51 views

Definite integral involving $\exp((-1/x) - x)$

I have a small integration problem. Just for context, this comes out of a bayesian exercise involving gamma functions. $$\int_{0}^{\infty} x^\alpha e^{-\left( \beta x^{-1} + \gamma x \right)} dx$$ ...
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84 views

What's the logic behind this step?

Let $f(y)$~$exp(\theta)$ if $y>0$, and $f(y)$ be $0$ elsewhere. Let $Y_1, Y_2, Y_3, Y_4$ be random samples from $f(y)$. Given $X=\sqrt{Y_1\cdot Y_2}$, the expected value of each observation is $E[...
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Is this a gamma distribution. What are the parameters [duplicate]

Suppose that $T_1,…,T_{10}$ are iid $Exp(λ)$ and that $S=\sum_{i=1}^{10}Ti$. In this case, S is a gamma distribution right? How would you write the parameters for this Gamma?
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1answer
72 views

Poisson process and compute some probabilities

Let $\{N(t) :t\geq0\}$ be a Poisson process with rate $\lambda$ and let $T_1,T_2,\dots$ be the arrival times. a)What is the probability that there are three arrivals in the time interval $[2,6]$and ...
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1answer
28 views

Gamma distribution with rate

I'm trying to solve this question: Let T be a random variable with Gamma(r=7, LAMBDA) distribution, where r is the shape parameter and LAMBDA the rate parameter. What is P(T > E[T])? I'm trying to ...
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2answers
61 views

Let X be a random variable with pdf given by $\beta^2xe^{-\frac{1}{2}\beta^2x^2}$

Let X be a random variable with pdf given by $\beta^2xe^{-\frac{1}{2}\beta^2x^2}$ where $\beta>0, x >0$ Find $E[X]$. I believe this is a function from the gamma distribution. Here is what I have ...
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14 views

pdf of random variable which is product of two gamma random variables and their summation.

I want to find PDF of random variable $Y$ which is product of two gamma random variables and their summation i.e., $Y=\sum_{k=1}^{M}h_1\cdot h_2$ where $h_1,h_2$ both are i.i.d gamma random variables. ...
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51 views

UMVU etimator for gamma distribution with unknown alpha

Given $X_1,X_2,...,X_n$ random samples from $gamma(α,β)$ with unknown $α$. What will be the "Uniform minimum variance unbaised" Estimator for $α$ and $β$. If $α$ is known then we know the ...
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1answer
38 views

Sum of inverse gamma random variables with unity parameters? (challenging integral)

I came across a problem on this site (which I now cannot find) asking about the distribution of sums of i.i.d. inverse gamma random variables for paramters $\alpha=\beta=1$. Defining $Z=X+Y$ with $X,Y\...
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12 views

Gamma transformation

I am trying to find the distribution of $X/(X+cY)$ where X~$Gamma(n,\lambda/n)$ and Y~$Gamma(n,c\lambda/n)$ where c is a constant. I have applied the transformation w=X/(X+cY) and z=X+cY so that x=wz ...
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24 views

How to obtain the PDF of product of two iid gamma random variables

Suppose I have two Gamma rvs $X1\sim G(m1,\Omega1)$ and $X2\sim G(m2,\Omega2)$ and both are i.i.d. Then how to derive the PDF of product of $X1\cdot X2$. Any help in this regard is highly appreciated.
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29 views

Compound Random Variable

May I ask a question regarding compound random variables, in which I want to obtain the distribution of $T$? $T=\sum_{i=1}^{N}t_i$, where $t_i$ are i.i.d exponential random variables with parameter $\...

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