generateExponential(lambda)

Overview

The generateExponential(lambda) function generates a random number that follows an Exponential distribution.

Parameters

Returns

Example

local x = StatBook.generateExponential(0.5)
print(x)  -- Output will vary

Mathematical Background

The function generates a random number $x$ that follows an Exponential distribution with rate parameter $\lambda$. The Exponential distribution is often used to model the time between events in a Poisson process.

To generate $x$, the function uses the following formula:

where $U$ is a uniformly distributed random number between 0 and 1.

The $\ln$ function represents the natural logarithm, and $\lambda$ is the rate parameter, which should be greater than zero.

x = \frac{d \times v}{\beta} ]

where $d = \alpha - \frac{1}{3}$, $v = (1 + c \times Z)^{3}$, and $c = \frac{1}{\sqrt{9 \times d}}$.

The reciprocal of $x$ from the Gamma distribution is then taken to generate the Inverse Gamma distributed number.