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(some links; expanded example a bit) 

==Underlying principle==
Let the unknown [[Parameter#Statistics_and_econometricsparameter]] of interest be <math>\mu</math>, and assume we have a [[statistic]] <math>m</math> such that the [[expected value]] of ''m'' is μ: <math>\mathbb{E}\left[m\right]=\mu</math>, i.e. ''m'' is an [[bias of an estimatorunbiased estimator]] for μ. Suppose we calculate another statistic <math>t</math> such that <math>\mathbb{E}\left[t\right]=\tau</math> is a known value. Then
:<math>m^\star = m + c\left(t\tau\right) \, </math>
is also
The [[variance]] of the resulting estimator <math>m^{\star}</math> is
:<math>\rho_{m,t}=\textrm{Corr}\left(m,t\right); \, </math>
In the case that <math>\textrm{Cov}\left(m,t\right)</math>, <math>\textrm{Var}\left(t\right)</math>, and/or <math>\rho_{
==Example==
We would like to estimate
:<math>I = \int_0^1 \frac{1}{1+x} \, \mathrm{d}x
▲The exact result is <math>I=\ln 2 \approx 0.69314718</math>. Using [[Monte Carlo integration]], this integral can be seen as the expected value of <math>f(U)</math>, where
:<math>f(x) = \frac{1}{1+x}</math>
and ''U'' follows a [[uniform distribution (continuous)uniform distribution]] [0, 1].
Using a sample of size
:<math>I \approx \frac{1}{n} \sum_i f(u_i); </math>
:<math>I \approx \frac{1}{n} \sum_i f(u_i)+c\left(\frac{1}{n}\sum_i g(u_i) 3/2\right). </math>
Using <math>n=1500</math> realizations and an estimated optimal coefficient <math> c^\star \approx 0.4773 </math> we obtain the following results
}
The variance was significantly reduced after using the control variates technique. (The exact result is <math>I=\ln 2 \approx 0.69314718</math>.)
==See also==
