Control variates: Difference between revisions

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and ''U'' follows a [[uniform distribution (continuous)|uniform distribution]] [0, 1].
Using a sample of size ''n'' denote the points in the sample as <math>u_1, \cdots, u_n</math>. Then the estimate is given by
:<math>I \approx \frac{1}{n} \sum_i f(u_i);. </math>
 
Now we introduce <math>g(U) = 1+U</math> as a control variate with a known expected value <math>\mathbb{E}\left[g\left(U\right)\right]=\int_0^1 (1+x) \, \mathrm{d}x=\tfrac{3}{2} </math> and combine the two into a new estimate