# Simulation and Random Variable Generation

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## Abstract

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:where This fact is sometimes referred to as the “Law of the Unconscious Statistician” (Allen 2006). Now suppose And therefore:So, if we randomly generated

$$ I={\displaystyle \underset{a}{\overset{b}{\int }}}f(x)dx $$

*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$$ E\left[f(U)\right]={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}}f(u)g(u)du $$

*U*is uniformly distributed between*a*and*b*. Then$$ g(u)=\frac{1}{b-a}\ \forall u,\ a\ \le u\ \le b $$

$$ I=\left(b-a\right)E\left(f(U)\right)=\left(b-a\right){\displaystyle \underset{a}{\overset{b}{\int }}}f(u)\frac{1}{b-a}du $$

*N*values of*U*,*u*_{1},*u*_{2}, …,*u*_{ N }, and computed## Keywords

Heat Transfer Apply Scientist Average Acceleration Uniform Probability Distribution Random Variable Generation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

- Allen, T. T. (2006).
*Introduction to engineering statistics and six sigma*. London: Springer.Google Scholar - Cryer, J. D. (1986).
*Time series analysis*. Boston: Duxbury Press.Google Scholar - Incropera, F. P., & De Witt, D. P. (1990).
*Introduction to heat transfer*(2nd ed.). New York: Wiley.Google Scholar - Law, A. M., & Kelton, W. D. (1982).
*Simulation modeling and analysis*. New York: McGraw-Hill.Google Scholar - Springer, M. D. (1979).
*The algebra of random variables*. New York: Wiley.Google Scholar

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