This method of common numbers produces good, but not overwhelming, variance reduction. The method of common random numbers (also known as the method of correlated sampling, the method of matched pairs, or the method of matched sampling) does not always work. It can backfire if the the engineer of the Monte Carlo simulation creates a negative, rather than positive, correlation between the two random variables

and

. Often, it is useful to choose

, which we do in this example.

Why does this method work?

Recall that, for example,

is a sequence of independent, identically distributed random variables. The variance of the first Monte Carlo method, when

is independent of

, is

.

Now consider the second Monte Carlo method, the method of common random numbers. The variance of this Monte Carlo method, when

is positively correlated to

, is

.

If we make the additional assumption that

and

are either: (a) both monotonically nondecreasing; or (b) both monotonically nonincreasing, then

,

and we see that

.

Notice that in both cases,

and

have the identical marginals—the method of common numbers only permits us to manipulate the joint distribution of

and

. Also, notice that in our example, both

and

are strictly monotonically increasing.

In this particular example, the variance reduction is always successful. However, notice that other measures of dispersion—like the range or the interquartile range—are not always reduced by the technique.

There are more powerful variance reduction techniques available, including antithetic variates, control variates, importance sampling, and stratified sampling. For more information on the method of common random numbers, see Sheldon Ross's textbook

*Stochastic Processes* or Paul Glasserman's book on Monte Carlo methods.