Simulation and Random Variable Generation

Originally, simulation meant using an electronic computer to generate pseudo-random numbers, uniformly distributed between 0 and 1 (Law and Kelton 1982). The single use of these numbers, called pseudo-random because they appear to have a uniform probability distribution, but in fact any sequence of them can be predicted exactly, was to perform numerical integration. Suppose an EAS wanted to compute a numerical approximation to an integral: I = ∫ a b f x d x $$ I={\displaystyle \underset{a}{\overset{b}{\int }}}f(x)dx $$ where f(x) is some fairly complicated and intractable function of x. Of course, nowadays there are many excellent numerical integration codes available. However, there was a time when such computing facilities were not easily obtained. It is true that if f(U) is a function of a random variable, U, and U has density function g(u), then E f U = ∫ − ∞ + ∞ f u g u d u $$ E\left[f(U)\right]={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}f(u)g(u)du $$ This fact is sometimes referred to as the “Law of the Unconscious Statistician” (Allen 2006). Now suppose U is uniformly distributed between a and b. Then g u = 1 b − a ∀ u , a ≤ u ≤ b $$ g(u)=\frac{1}{b-a}\ \forall u,\ a\ \le u\ \le b $$ And therefore: I = b − a E f U = b − a ∫ a b f u 1 b − a d u $$ I=\left(b-a\right)E\left(f(U)\right)=\left(b-a\right){\displaystyle \underset{a}{\overset{b}{\int }}}f(u)\frac{1}{b-a}du $$ So, if we randomly generated N values of U, u 1, u 2, …, u N , and computed