A random variable X has P{X 0} 0.9. Given X  0, it has a uniform distribution between 5 and 15. Thus, E(X) 1.

Obtaining uniform random numbers as instructed at the beginning of the Problems section, use simulation to estimate E(X).

(a) Estimate E(X) by generating five random observations from the distribution of X and then calculating the sample average.

(This is the crude Monte Carlo technique.)

(b) Estimate E(X) by using stratified sampling with two strata—

0 F(x) 0.9 and 0.9  F(x) 1—with 1 and 4 observations, respectively.