# Why?

## November 28, 2010

### Random variable generation (Pt 1 of 3)

Filed under: AMCMC, R — Tags: , , , , — csgillespie @ 7:35 pm

As I mentioned in a recent post, I’ve just received a copy of Advanced Markov Chain Monte Carlo Methods. Chapter 1.4 in the book (very quickly) covers random variable generation.

## Inverse CDF Method

A standard algorithm for generating random numbers is the inverse cdf method. The continuous version of the algorithm is as follows:

1. Generate a uniform random variable $U$

2. Compute and return $X = F^{-1}(U)$

where $F^{-1}(\cdot)$ is the inverse of the CDF. Well known examples of this method are the exponential distribution and the Box-Muller transform.

## Example: Logistic distribution

I teach this algorithm in one of my classes and I’m always on the look-out for new examples. Something that escaped my notice is that it is easy to generate RN’s using this technique from the Logistic distribution. This distribution has CDF

$\displaystyle F(x; \mu, s) = \frac{1}{1 + \exp(-(x-\mu)/s)}$
and so we can generate a random number from the logistic distribution using the following formula:
$\displaystyle X = \mu + s \log\left(\frac{U}{1-U}\right)$

Which is easily converted to R code:
 myRLogistic = function(mu, s){   u = runif(1)   return(mu + s log(u/(1-u))) }