Questions tagged [gamma-distribution]

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

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15
votes
1answer
27k views

What is the relationship between poisson, gamma, and exponential distribution?

I'm having a hard time understanding the intuitive relationship between these three distributions. I thought that poisson is what you get when you sum n number of exponentially distributed variables, ...
6
votes
3answers
103 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}(...
6
votes
2answers
2k views

The intuition behind gamma distribution

What is the intuition behind gamma distribution? For instance, I understand how to "construct" Gaussian distribution. This is my intuition: Bernoulli distribution - which is simple concept A ...
5
votes
1answer
2k views

Distribution of weighted sum of Bernoulli RVs

Let $x_1,...,x_m$ be drawn from independent Bernoulli distributions with parameters $p_1,...,p_m$. I'm interested in distribution of $t=\sum_i a_ix_i,~a_i\in \mathbb{R}$ $m$ is not large so I can ...
5
votes
2answers
140 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 ...
5
votes
1answer
4k views

Derivation of Distribution Function (CDF) of Gamma Distribution using Poisson Process

I found the following result on Wikipedia relating to the CDF of the Gamma Distribution when the shape parameter is an integer. (Note: there is a slight difference on how I have defined the scale ...
5
votes
1answer
145 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(\...
4
votes
2answers
99 views

Compute some expectations involving beta and gamma distributions

a) Let $X \sim {\text {Beta}}(a,b)$, with density $f(x) = \frac{x^{a-1}(1-x)^{b-1}} {B (a,b)}$. Find the expectation of $X \ln X$ and $\frac{1}{X+1}.$ b) Let $Y \sim {\text {Gamma}}(\alpha,\beta)$, ...
4
votes
2answers
139 views

Joint distribution of $X_1/(X_1+X_2)$ and $X_2/(X_1+X_2)$ for independent exponential random variables $X_1$ and $X_2$

If $X_i$, $i=1,2$, are independent $gamma(\alpha_i,1)$ random variables. Find the joint distribution of $X_1/(X_1+X_2)$ and $X_2/(X_1+X_2)$. I am trying to use transformation method to solve it. Let ...
4
votes
0answers
232 views

Finding Bayes estimator with inverse-gamma prior and uniform likelihood.

Consider an i.i.d. sample of data from a population given by $$Y_i|\Psi \sim \text{Unif}(0, \Psi), \quad i = 1,2,\ldots, n,$$ with $\Psi$ having a prior distribution given by $$\Psi \sim \text{Inv-}\...
4
votes
0answers
830 views

Central limit theorem for positive random variables

Let $X_1, \ldots, X_n$ be a set of $n$ i.i.d. samples of a non-negative random variable $X$ with $\mathrm{E}(X)=\mu$ and $\mathrm{Var}(X)=\sigma^2$. By the central limit theorem, the sample mean $\hat{...
4
votes
0answers
2k views

Problem with the expectation of a maximum of independent gamma distributed random variables

Having a problem with the expectation of the maximum among $n$ independent random variables $ X_1, X_2 \dots X_n$ all ~ the same class of distributions but not necessarily the same mean and other ...
3
votes
1answer
2k views

Show that the gamma density integrates to 1.

$$\int_{-\infty}^{\infty} g(t)dt = \int_{0}^{\infty} \frac{\lambda^{\alpha}}{\Gamma(\alpha)} t^{\alpha - 1} e^{-\lambda t} dt = \frac{\lambda}{\Gamma(\alpha)} \int_{0}^{\infty} (\lambda t)^{\alpha - 1}...
3
votes
3answers
1k views

How to show that $f(x; \alpha, \beta) = \frac{1}{\beta^{\alpha} \Gamma(\alpha)}x^{\alpha-1}e^{-x/\beta}$is a pdf?

I have some problems trying to prove the following problem: A continuous random variable $X$ is said to have a gamma distribution with parameters $\alpha > 0$ and $\beta > 0$ if it has a pdf ...
3
votes
1answer
133 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 ...
3
votes
2answers
3k views

Find the Laplace transform of the Gamma pdf

Per wikipedia the Laplace transform of the gamma distribution is $$L_X(s) = (1+\theta s)^{-k} = \frac{\beta^\alpha}{(s+\beta)^\alpha}$$ As an exercise I would like to show this.The definition I have ...
3
votes
2answers
121 views

What is $\int_0^{\infty}e^{-x^{1.5}+\theta x}dx=?$

Is there a way to obtain an expression for $$\int_0^{\infty}e^{-x^{1.5}+\theta x}dx=?$$ If $\theta=0$, we know the above is the same as $\frac{\Gamma(2/1.5)}{\Gamma(1/1.5)}$ from a generalized Gamma ...
3
votes
1answer
233 views

Probability that one station becomes empty before another.

Question: There are 2 stations A and B in series having i and j customers respectively. Customers after being served at station A are routed to station B. The service time of each of the queues are ...
3
votes
2answers
285 views

Conditional distribution of $X$ given that $X+Y=2$

I have $X$ and $Y$ independent $\Gamma(2,a)$-distributed random variables. I am trying to find conditional distribution of $X$ given that $X+Y=2$. My solution: set $U = X+Y, V = X.$ Inversion yields $...
3
votes
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 ...
3
votes
1answer
396 views

Chi square and gamma distribution.

I have studied the intuiton of the gamma distribution and have understood the following: Let us suppose that we want to study the probability of waiting time until the $\alpha$-th event occurs. Let $...
3
votes
1answer
62 views

customer service time problem

In a disc shop two employees are working. When we get inside the shop, we see that the two employees are already serving two customers (one customer for each employee), with the service time being a ...
3
votes
2answers
244 views

Writing the pdf for a Gamma Distribution

Let $X_1, \dots, X_5$ be 5 independent variables from the exponential distribution with the mean $2$. a) The pdf of $T=X_1 + \dots + X_5$ b) $P(T > 5)$ c) $E(T/5)$ Having a little trouble ...
3
votes
1answer
100 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} \ \ \ \ \ \...
3
votes
1answer
371 views

sum of two independent scaled noncentral $\chi$-squared random variables

I want to analyze or approximate a random variable that is a sum of two scaled independent non central $\chi$-squared random variables with the same degrees of freedom. For example, $$X = X_1 + a X_2$...
3
votes
1answer
32 views

Estimating Gamma PDF parameters from data with negative increments

Say we have collected data, and from a physical perspective we know that the collected data should increase positively with time. However the data looks more like this: This data shown in the figure ...
3
votes
0answers
43 views

Survival Gamma Function

I answered a question in my probability test but I think it's wrong! Could you tell me with my solution is right? The question is: Find $$\lim_{n\to\infty}\int_{2n}^\infty\frac{1}{2^n(n-1)!}t^{n-1}e^...
3
votes
0answers
152 views

Invert a somewhat tricky characteristic function to find density function

I am interested in find the probability density function corresponding to the characteristic function $\phi(t) = \left(\frac{1 - i b t}{1 - i t}\right)^c$ where $c > 1$ and and $0< b < 1$. ...
2
votes
1answer
987 views

Mean of gamma distribution.

So I was trying to prove the mean result of gamma distribution which is $\frac{\alpha}{\lambda}$. My attempt, $E(X)=\int_{0}^{\infty }x f(x)dx$ $=\int_{0}^{\infty } \frac{\lambda^{\alpha}}{\Gamma (...
2
votes
5answers
67 views

How to integrate using known distributions

I'm having trouble figuring this integration out using known distributions. I don't know which distribution to use to solve this problem. It looks like a gamma to me. $$\int_{0}^{\infty} x^{3}e^{-x^2}...
2
votes
2answers
144 views

approximation for $\Gamma (\alpha) / \Gamma (\beta) $ where $\alpha$ and $\beta$ are arbitrary numbers in $R^{+}$

I am working on implementation of a machine learning method that in part of the algorithm I need to calculate the value of $\Gamma (\alpha) / \Gamma (\beta) $. $\alpha$ and $\beta$ are quite large ...
2
votes
2answers
564 views

Poisson Processes with Gamma Arrivals

I think the title is the best description I can give of my problem (but I'm not 100% sure - the problem set-up has me very confused). So, given a sequence of i.i.d. Gamma RV having parameters 3, $\...
2
votes
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 ...
2
votes
1answer
5k views

Determine the mode of the gamma distribution with parameters $\alpha$ and $\beta$

How do you determine the mode of a gamma distribution with parameters $\alpha$ and $\beta$ ? Without looking on Wikipedia.
2
votes
1answer
6k views

How to find the mode and median of a Gamma distribution? [closed]

A random variable has Gamma distribution with mean of $10$ and standard deviation of $5$. The mode and median are to be found. I realize that this means that $\alpha$ and $\beta$ are both $\sqrt{5}$....
2
votes
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 $$\...
2
votes
2answers
63 views

Finding the pdf of a random variable generating from another random variable with defined pdf

Initially, there is a random variable $X$ (non-negative) with distribution function: $$P(x) = \lambda e^{-\lambda x}$$ Now we randomly generate $X$ and, then, form sets of $N$ values of this variable: ...
2
votes
2answers
48 views

Inconsistent interpretations for the second parameter of the Gamma distribution

The following question is motivated by material from pages 358-364 of Blitzstein and Hwang's Introduction to Probability. (NB: I have modified the book's notation in various places to make my ...
2
votes
1answer
225 views

Distribution of $X$ Poisson with parameter $\lambda$ if the distribution of $\lambda$ is Gamma $(2,2)$

This is from Exercise 6.15 in Canavos' Applied Probability and Statistical Methods. I cannot get my result to match answer given at the end of the book. Given a Gamma distribution with shape and ...
2
votes
1answer
144 views

Relationship between the incomplete gamma function of 2a and a

If the gamma function is given by $$\Gamma(\alpha) = \int_0^{+\infty}t^{\alpha-1}e^{-t}\text dt$$ and the lower incomplete gamma function by $$\gamma(\alpha,x) = \int_{0}^{x}t^{\alpha-1}e^{-t}\text ...
2
votes
1answer
63 views

Probability that $\operatorname{Erlang}(2,\mu _2)$ is greater than $\exp(\mu _1)$

I'm trying to work out that, given that $X\sim \exp(\mu_1)$ and $Y\sim \operatorname{Erlang}(2,\mu _2)$, what is $\mathbb{P}(X<Y)$? So far I have: $$\mathbb{P}(X<Y)=\int_0^{\infty}\mathbb{P}(X&...
2
votes
1answer
47 views

Name of the distribution with density $P(x) e^{-x/\theta}$, where $P$ is a polynomial with positive coefficients.

The Gamma distribution of shape $k$ and scale $\theta$ has density $\frac1{\Gamma(k)\theta(k)} x^{k-1} e^{-x/\theta}$. Consider the more general distribution with density (up to a normalizing constant)...
2
votes
1answer
372 views

Differential entropy of $\Gamma$

Let $X \sim Gamma(\alpha,\beta)$ be gamma distributed random variable with probability distribution function $$ f_{X}(x)=\frac{\beta^{\alpha}x^{\alpha-1}e^{-\beta x}}{\Gamma(\alpha)},\;x>0 $$ ...
2
votes
1answer
250 views

Give necessary and sufficient conditions so the sum of random variables converges almost surely

$\{X_k\}_{k}$ are independent random variables on the probability space $(\Omega, \mathcal F, P)$ and $X_k$ has gamma density $f_k(x)$ where $f_k(x)=\dfrac{x^{a_x-1}e^{-x}}{\Gamma(a_k)}$ where $x,a_k &...
2
votes
1answer
458 views

Let $X|Y = y\sim\text{Poisson}(y)$ and $Y\sim\text{Gamma}(\alpha, \lambda)$. Find $f_X(x)$.

Question: Let $X|Y = y\sim\text{Poisson}(y)$ and $Y\sim\text{Gamma}(\alpha, \lambda)$. Find join density $f_{X,Y}(x,y)$ and find the probability density function of $X$ (simplify until there are no ...
2
votes
2answers
350 views

chi squared distribution of independent normal distributions that are not standard normal

I've been working on the following problem. I'm a bit confused about some of the specifics of how to arrive at the correct answer. I hope someone here could point me in the right direction: A dart ...
2
votes
1answer
37 views

Unit Measure Axiom for the Gamma Distribution

I'm studying basic probability and in my lecture notes, it shows how the Gamma function results from the convolution of two exponential random variables. To introduce the gamma function it shows that ...
2
votes
1answer
81 views

Evaluate $\lim\limits_{\alpha \to \infty} e^{-\frac{t}{\sqrt{\alpha}}}(1-\frac{t}{\sqrt{\alpha}})^{-\alpha}$ [duplicate]

How does one show $$\lim_{\alpha \to \infty} e^{-t\sqrt{\alpha}}\left(1-\frac{t}{\sqrt{\alpha}}\right)^{-\alpha} = e^{t^2 / 2}?$$ Not homework, this is from this proof that the gamma distribution ...
2
votes
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,...
2
votes
1answer
81 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 ...