Consider the probability distribution whose probability density function is f(x)

The problem is to perform a simulated experiment, with the help of variance-reducing techniques, for estimating the mean of this distribution. To provide a standard of comparison, also derive the mean analytically.

For each of the following cases, use the same 10 uniform random numbers (obtained as instructed at the beginning of the Problems section) to generate random observations, and calculate the resulting estimate of the mean.

(a) Use the crude Monte Carlo technique.

(b) Use stratified sampling with three strata—0 F(x) 0.6, 0.6  F(x) 0.9, and 0.9  F(x) 1—with 3, 3, and 4 observations, respectively.

(c) Use the method of complementary random numbers.