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2.4.7 Monte Carlo Integration (Our First Application of Monte Carlo)

We would like to evaluate the following definite integral,

where we assume that is real-valued on . Figure 9 depicts a typical integral to be evaluated.

Figure 9: Monte Carlo Integration.

The idea is to manipulate the definite integral into a form that can be solved by Monte Carlo. To do this, we define the following function on ,

and insert into Eq. (64) to obtain the following expression for the integral I:

Note that can be viewed as a uniform pdf on the interval , as depicted in Figure 9. Given that is a pdf, we observe that the integral on the right hand side of Eq. (66) is simply the expectation value for :

Figure 10: Uniform pdf on [a,b].