Simulation and Random Variable Generation

  • Scott A. Pardo


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:
$$ I={\displaystyle \underset{a}{\overset{b}{\int }}}f(x)dx $$
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
$$ E\left[f(U)\right]={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}f(u)g(u)du $$
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
$$ g(u)=\frac{1}{b-a}\ \forall u,\ a\ \le u\ \le b $$
And therefore:
$$ I=\left(b-a\right)E\left(f(U)\right)=\left(b-a\right){\displaystyle \underset{a}{\overset{b}{\int }}}f(u)\frac{1}{b-a}du $$
So, if we randomly generated N values of U, u1, u2, …, u N , and computed


Heat Transfer Apply Scientist Average Acceleration Uniform Probability Distribution Random Variable Generation 
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  4. Law, A. M., & Kelton, W. D. (1982). Simulation modeling and analysis. New York: McGraw-Hill.Google Scholar
  5. Springer, M. D. (1979). The algebra of random variables. New York: Wiley.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Scott A. Pardo
    • 1
  1. 1.Ascensia Diabetes CareParsippanyUSA

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