a control variate
Imagine measuring the weight of a captain by weighing him together with his dog, when you already know the dog's exact weight. The combined measurement is noisy, but subtracting the dog's KNOWN weight removes a chunk of the variation and leaves a sharper estimate of the captain. A control variate does the same to a Monte Carlo estimate: it borrows a related quantity whose true mean you happen to know exactly, and uses the error you make ON that known quantity to correct your estimate of the unknown one.
Here is the mechanism. You want E[f], the mean of f, by averaging samples. Pick a second quantity g that is correlated with f and whose exact mean E[g] you know. For each sample form the adjusted estimator f(x_i) - c * (g(x_i) - E[g]), and average it. Because g - E[g] has mean zero, subtracting any multiple c of it leaves the estimator UNBIASED — its average is still E[f]. But it changes the variance: choosing c well cancels the part of f's fluctuation that g predicts. The optimal coefficient is c* = Cov(f, g) / Var(g), and with it the variance drops by the factor (1 - rho^2), where rho is the correlation between f and g. So a control variate 90% correlated with f (rho = 0.9) removes about 81% of the variance — like quintupling your sample count for free.
Control variates are a favourite in computational finance (price a hard option using a similar option with a known closed-form price as the control) and in any simulation where a simplified, analytically solvable version of the problem sits nearby. The honest points: you must actually KNOW E[g] exactly (an approximation reintroduces bias), the benefit is only as large as the correlation rho, and in practice c* is estimated from the same samples, which introduces a small bias unless handled carefully (e.g. by splitting the data). A control g that is uncorrelated with f does nothing; a strongly correlated one is gold.
Estimate E[e^U] for U uniform on [0,1] (true value e - 1 = 1.7183). Use g = U as the control, since E[U] = 0.5 is known exactly and e^U correlates strongly with U. The adjusted estimator e^(U_i) - c*(U_i - 0.5) with the optimal c removes most of the variance, giving a far tighter estimate at the same N.
Subtract a correlated quantity with known mean to cancel noise — for free and unbiased.
The whole method hinges on knowing E[g] EXACTLY; an approximate mean turns variance reduction into bias. The variance only drops by 1 - rho^2, so a weakly correlated control barely helps, and estimating the coefficient c from the same data adds a small bias unless you split the samples.