Questions tagged [gamma-distribution]

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

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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 ...
6
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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 ...
4
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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 $Y \sim {\text {Gamma}}(\alpha,\beta)$, ...
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}(...
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(\...
2
votes
1answer
984 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 (...
1
vote
1answer
815 views

Scale versus rate parameterization in exponential and gamma distributions

In some cases, I've seen the PDFs of the exponential and gamma distribution presented as: $$X\sim Exp(\lambda) \rightarrow f(x) = \lambda e^{-\lambda x}$$ $$X\sim Gamma(\alpha, \lambda) \rightarrow f(...
0
votes
1answer
123 views

Behavior of Gamma Distribution over time

I need to prove that the age behavior of the Gamma Distribution with probability density function $$ f(x)=\frac{\lambda^{\alpha}x^{\alpha-1}e^{-\lambda x}}{\Gamma(\alpha)}$$ For $x\geq 0$, $\lambda, \...
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
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1answer
786 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
1answer
3k views

PDF of the product of two independent Gamma random variables

I am trying to find out the density of the product $XY$ of two independent Gamma random variables $X \sim \mathrm{Gamma}(k_1, \theta_1)$ and $Y \sim \mathrm{Gamma}(k_2, \theta_2)$, where $k_i$'s are ...
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
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 $\sqrt{5}$....
1
vote
1answer
150 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 ) \...
1
vote
3answers
1k views

Product of Uniform Distribution and $\Gamma(2,1)$ Distribution

I ran into an old exercise but I seem to have messed up somehow. Can you tell me what went wrong? Let $U \sim \mathrm{Unif}(0,1)$ and $V \sim \Gamma(2,1)$ with $U,V$ independent. Show that $UV$ has ...
1
vote
1answer
2k views

Bootstrap estimation of the 95% confidence intervals for the 95% quantile for gamma distribution

I cant find any where information or algorithm how to apply in steps the bootstrap procedure to estimate the 95% confidence intervals for the 95% quantile from a random sample. Does anyone knows how ...
0
votes
1answer
1k views

How to scale a gamma distribution [closed]

I am trying to sample from a gamma distribution using transform sampling. All the example I seen as in the Wikipedia, is on the range from 0 to 20. How can I scale or extend the distribution over say ...
0
votes
1answer
2k views

Sum of gamma and normal random variable

If $X$ has an exponential distribution and $Y$ is normally distributed random variable, then what is the distribution of $Z=X+Y$?