## Quick Reference

A method of reducing the variance of an estimate obtained via simulation. Suppose, for example, that we wish to estimate the quantity *I*, given by , where g is a given function. An obvious method is to generate uniform pseudo-random numbers *u*_{1}, *u*_{2},…, *u** _{n}*, in the interval (0, 1). Writing

*u*

_{1},

*u*

_{2},…,

*u*

*for the corresponding random variables, the estimator,*

_{n}*I*

_{1}, is . Importance sampling makes a more representative choice of values: suppose that f is a probability density function that resembles g in its general shape, and let F be the corresponding distribution function. Instead of working with

*u*

_{1},

*u*

_{2},…,

*u*

*, we work with ν*

_{n}_{1}, ν

_{2},…, ν

*, where F(ν*

_{n}*)=*

_{j}*u*

*. The resulting estimator,*

_{j}*I*

_{2}, given by , will have a smaller variance than

*I*

_{1}.

*Subjects:*
Probability and Statistics.