Links: Probability Functions for Random Variables

General Remarks

Let $X$ be a random variable and let $F$ be the cdf, with an inverse $F^{-1}$, and $U\sim \mathrm{unif}(0,1)$, then

• $F(X)\stackrel{d}{=}U$
• $F^{-1}(U)\stackrel{d}{=}X$

Meaning if we can calculate the inverse of a cdf, then we can model the random variable using just an uniform distribution

Examples

there are quite a few continuous distributions:

• Uniform Continuous Distribution
• Exponential Distribution
• Gamma Distribution
• Beta Distribution
• Weibull Distribution
• Normal Distribution
• Lognormal Distribution
• Chi-squared Distribution
• Student's t-distribution
• F-distribution
  Multivariate distributions:
• Multivariate Uniform Distribution
• Multivariate Normal Distribution