## Quick Reference

A method for generating values of a continuous random variable for use in a simulation. Suppose that the random variable *X*, which takes values in the interval (*a*, *b*), has probability density function f. Denote the maximum value of f(*x*) by *M*. Let *u* and *ν* be two random numbers uniformly distributed in the interval (0, 1). Write *r*=*a*+(*b*−*a*)*u* and *s*=*Mν*, so that *r* and *s* are uniformly distributed on (*a*, *b*) and (0, *M*), respectively. Calculate f(*r*). If f(*r*)>*s* then *r* is accepted as a value of *X*. Otherwise, it is rejected and a new pair of values is taken for *u* and *ν*.

**Acceptance–rejection algorithm.** Uniform random numbers are generated in the intervals (*a*, *b*) and (0, *M*). If the point generated lies between the graph of f(*x*) and the *x*-axis, then the value of *X* is accepted.

*Subjects:*
Probability and Statistics.