generateGamma(alpha, beta)

Overview

The generateGamma(alpha, beta) function generates a random number that follows a Gamma distribution with the given shape parameter $\alpha$ and scale parameter $\beta$. The function uses the Marsaglia and Tsang method for this purpose.

Parameters

Parameter	Type	Description	Default
alpha	Number	The shape parameter of the Gamma distribution.	-
beta	Number	The scale parameter of the Gamma distribution.	-

Returns

Return	Type	Description
x	Number	A random number from a Gamma distribution with parameters $\alpha$ and $\beta$.

Example

local alpha = 2
local beta = 1
local x = StatBook.generateGamma(alpha, beta)
print(x)  -- Output will vary

Mathematical Background

The function utilizes the Marsaglia and Tsang method to generate a random number $x$ that follows a Gamma distribution with shape parameter $\alpha$ and scale parameter $\beta$.

For $\alpha < 1$, the function uses a recursive approach to find $x$ such that:

[ x = \text{GenerateGamma}(\alpha + 1, \beta) \times u ^ {\frac{1}{\alpha}} ] Where $u$ is a uniformly distributed random number between 0 and 1.

For $\alpha \geq 1$, the function uses a while loop to generate $x$ as:

[ x = \frac{d \times v}{\beta} ] Where $d = \alpha - \frac{1}{3}$ and $v$ is calculated using a standard normally distributed random number $Z$ generated by Box-Muller transform as:

[ v = (1 + c \times Z)^3 ] And $c = \frac{1}{\sqrt{9 \times d}}$.

The function repeats this process until a suitable $x$ is found.