F f2 N 1 w

where the numerator under the square root may be estimated from the taken sample. One can thus reduce the error by either increasing N or reducing the mean standard deviations of f.

The above integration method, although convergent, might not be very efficient if the function has large variations in the interval. Consider for instance f (x) = e10x in the interval [0,1); clearly a large number of

points and function evaluations will be "wasted" near x=0 where the function is much smaller (by more than 4 orders of magnitude) compared to points near x=1. It would be better to sample the interval in a nonuniform fashion so as to have more points near x=1 and fewer points near x=0. This is what importance sampling can do. We choose a non-uniform distribution function P( x) according to which the points xi will be sampled. The integral can also be written in the following form:

0 0

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