Question

I have the following problem, and am having trouble understanding part of the equation:

Monte Carlo methods to estimate an integral is basically, take a lot of random samples and determined a weighted average. For example, the integral of f(x) can be estimated from N independent random samples x_r by

alt text http://www.goftam.com/images/area.gif

for a uniform probability distribution of xr in the range [x1, x2]. Since each function evaluation f(xr) is independent, it is easy to distribute this work over a set of processes.

What I don’t understand is what f(x_r) is supposed to do? Does it feed back into the same equation? Wouldn’t that be an infinite loop?

Answer 1

Your goal is to compute the integral of f from x1 to x2. For example, you may wish to compute the integral of sin(x) from 0 to pi.

Using Monte Carlo integration, you can approximate this by sampling random points in the interval [x1,x2] and evaluating f at those points. Perhaps you’d like to call this MonteCarloIntegrate( f, x1, x2 ).

So no, MonteCarloIntegrate does not ‘feed back’ into itself. It calls a function f, the function you are trying to numerically integrate, e.g. sin.