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].