Monte Carlo Simulation

Adapted From Law and Kelton

What is Monte Carlo Simulation? A scheme employing random

numbers, U(0,1) random variates, which is used to solving certain stochastic or deterministic problems where the passage of time does not play an important role. It is rather static than dynamic.

Example Ex. Suppose that we want to

evaluate this integral: 

Where g(x) is a real valued function that is not integrable

I g x dxa

b

( )

To use simulation, let Y be that random variable (b-a)g(X), where X is a continuous random variable distributed uniformly

on [a,b]. Thus,

= I

E Y E b a g X( ) [( ) ( )] ( ) [ ( )]b a E g X

( ) ( ) ( )b a g x f x dxx

a

b

( )

( )

( )b a

g x dx

b aa

b

But, E(Y) can be estimated using the concept of the sample mean . Thus,

Y nY

nb a

g X

n

ii

n

ii

n

( ) ( )( )

1 1

Ex. Evaluate (Answer is 2).sin xdx0

Y n( )

n 10 20 40 80 160

2.213 1.951 1.948 1.989 1.993