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)))

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