Generating Exponential Random Variables from Uniforms

1. Goal

Generate a random variable $X\sim \exp (\lambda )$ with CDF:

$$F(x)=1-e^{-\lambda x}.$$

2. Inverse Transform

Let

$$u=F(x)=1-e^{-\lambda x}.$$

Solving for $x$ gives:

$$x=-\frac{1}{\lambda }\log (1-u).$$

Equivalently:

$$\boxed{X=-\frac{1}{\lambda }\log (u)}.$$

3. Code

% set up the parameters.
lam = 2;
 
% generate the rv's
uni = rand(1, n);
X = -log(uni) / lam;

Related Notes

• Generating Continuous Random Variables
• Change of Variables for Probability Distributions