Questions tagged [gamma-distribution]

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

A random variable $X$ that is gamma-distributed with shape $k$ and scale $\theta$ is denoted by $$X \sim \Gamma(k, \theta) \equiv \textrm{Gamma}(k, \theta)$$

The probability density function using the shape-scale parametrization is $$f(x;k,\theta) = \frac{x^{k-1}e^{-\frac{x}{\theta}}}{\theta^k\Gamma(k)} \quad \text{ for } x > 0 \text{ and } k, \theta > 0$$

Here $\Gamma(k)$ is the gamma function evaluated at $k$.

The cumulative distribution function is the regularized gamma function: $$F(x;k,\theta) = \int_0^x f(u;k,\theta)\,du = \frac{\gamma\left(k, \frac{x}{\theta}\right)}{\Gamma(k)}$$

where $\gamma(k, x/ \theta)$ is the lower incomplete gamma function.

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Sum of independent Gamma distributions is a Gamma distribution

If $X\sim \Gamma(a_1,b)$ and $Y \sim \Gamma(a_2,b)$, I need to prove $X+Y\sim\Gamma(a_1+a_2,b)$ if $X$ and $Y$ are independent. I am trying to apply formula for independence integral and just trying to multiply the gamma function but stuck ?
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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, but if seems that gamma is the same...Could…
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Gamma Distribution out of sum of exponential random variables

I have a sequence $T_1,T_2,\ldots$ of independent exponential random variables with paramter $\lambda$. I take the sum $S=\sum_{i=1}^n T_i$ and now I would like to calculate the probability density function. Well, I know that $P(T_i>t)=e^{-\lambda…
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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 sequence of Bernoulli trials is a Binomial distribution.…
Andreo
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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 not use central limit theorems. I have the…
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Method of moments with a Gamma distribution

I'm more so confused on a specific step in obtaining the MOM than completely obtaining the MOM: Given a random sample of $ Y_1 , Y_2,..., Y_i$ ~ $ Gamma (\alpha , \beta)$ find the MOM So I found the population and sample moments $u_1^{'}= \alpha…
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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 the shape parameters and $\theta_i$'s are the…
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Insurance company with claims following a Poisson Process. Calculate the probability that the capital is always positive throughout the first 4 days.

Suppose that claims are made to an insurance company according to a Poisson process with rate $10$ per day. The amount of a claim is a random variable that has an exponential distribution with mean $1,000$ dollars. The insurance company receives…
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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 $X = YZ \sim \operatorname{Gamma}(a, 1)$? $f_X(x) =…
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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 $$ \frac{\beta^{\alpha}}{\Gamma(\alpha)}\int_0^\infty g(x)x^{\alpha-1}e^{-\beta x}\,\mathrm dx=\left(\frac{\alpha}{\beta}-\mu\right)^{-1}, $$ where…
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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 integration by parts and then call to special functions, but…
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Probability of a Gamma r.v. greater than another.

I saw an interesting formula at: https://stats.stackexchange.com/questions/264861/probability-of-gamma-greater-than-exponential but I didn't know how to derive it. The question is: Let $X,Y$ be independent r.v.s and…
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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 parameter and how it is given on the Wikipedia…
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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 $\hat{\mu}_n=n^{-1}\sum_{i=1}^n$ converges in…
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Convex combination of Dirichlet random variables

For positive integer $k$, let $(X_1,\ldots,X_k)\sim\mathrm{Dir}(\alpha_1,\ldots,\alpha_k)$ be a probability distribution over $k$ items drawn from a $k$-component Dirichlet distribution and $p=(p_1,\ldots,p_k)$ be another fixed distribution. What is…
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