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

We now draw samples from the pdf , and for each we will evaluate and form the average G,

But Eq. (62) states the expectation value for the average of N samples is the expectation value for , , hence

Thus we can estimate the true value of the integral I on by taking the average of N observations of the integrand, with the r.v. x sampled uniformly over the interval . For now, this implies that the interval is finite, since an infinite interval cannot have a uniform pdf. We will see later that infinite ranges of integration can be accommodated with more sophisticated techniques.

Recall that Eq. (63) related the true variance in the average G to the true variance in g,

Although we do not know , since it is a property of the pdf and the real function , it is a constant. Furthermore, if we associate the error in our estimate of the integral I with the standard deviation, then we might expect the error in the estimate of I to decrease by the factor . This will be shown more rigorously later when we consider the Central Limit Theorem, but now we are arguing on the basis of the functional form of and a hazy correspondence of standard deviation with ``error''. What we are missing is a way to estimate , as we were able to estimate with G.