Editing VEGAS algorithm (section)

==Sampling method==
{{Further|Importance sampling}}
In general, if the Monte Carlo integral of <math>f</math> over a volume <math>\Omega</math> is sampled with points distributed according to a probability distribution described by the function <math>g,</math> we obtain an estimate <math>\mathrm{E}_g(f; N),</math>

:<math>\mathrm{E}_g(f; N) = {1 \over N } \sum_i^N { f(x_i)} / g(x_i) .</math>

The [[variance]] of the new estimate is then

:<math>\mathrm{Var}_g(f; N) = \mathrm{Var}(f/g; N)</math>

where <math>\mathrm{Var}(f;N)</math> is the variance of the original estimate, <math>\mathrm{Var}(f; N) = \mathrm{E}(f^2; N) - (\mathrm{E}(f; N))^2.</math>

If the probability distribution is chosen as <math>g = |f|/\textstyle \int_\Omega |f(x)|dx </math> then it can be shown that the variance <math>\mathrm{Var}_g(f; N)</math> vanishes, and the error in the estimate will be zero. In practice it is not possible to sample from the exact distribution g for an arbitrary function, so importance sampling algorithms aim to produce efficient approximations to the desired distribution.