Editing Mean squared error (section)

===Mean===
Suppose we have a random sample of size <math>n</math> from a population, <math>X_1,\dots,X_n</math>. Suppose the sample units were chosen [[Sampling with replacement|with replacement]]. That is, the <math>n</math> units are selected one at a time, and previously selected units are still eligible for selection for all <math>n</math> draws. The usual estimator for the population mean <math>\mu</math> is the sample average

:<math>\overline{X}=\frac{1}{n}\sum_{i=1}^n X_i </math>

which has an expected value equal to the true mean <math>\mu</math> (so it is unbiased) and a mean squared error of

:<math>\operatorname{MSE}\left(\overline{X}\right)=\operatorname{E}\left[\left(\overline{X}-\mu\right)^2\right]=\left(\frac\sigma{\sqrt n}\right)^2= \frac{\sigma^2}{n}</math>

where <math>\sigma^2</math> is the [[Sample variance#Population variance|population variance]].

For a [[Gaussian distribution]] this is the [[best unbiased estimator]] of the population mean, that is the one with the lowest MSE (and hence variance) among all unbiased estimators. One can check that the MSE above equals the inverse of the [[Fisher information]] (see [[Cramér–Rao bound]]). But the same sample mean is not the best estimator of the population mean, say, for a [[Uniform distribution (continuous)|uniform distribution]].