# Questions tagged [gamma-distribution]

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

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### 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 ...
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### 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 ...
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### 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 $Y \sim {\text {Gamma}}(\alpha,\beta)$, ...
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### 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.
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### 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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### 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 ...
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### 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 ...
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### 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 $\sqrt{5}$....
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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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### 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 ...
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### 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 ...
If $X$ has an exponential distribution and $Y$ is normally distributed random variable, then what is the distribution of $Z=X+Y$?