Editing Weight function (section)

=== Statistics ===
Weighted means are commonly used in [[statistics]] to compensate for the presence of [[Bias_(statistics)|bias]].  For a quantity <math>f</math> measured multiple independent times <math>f_i</math> with [[variance]] <math>\sigma^2_i</math>, the best estimate of the signal is obtained  by averaging all the measurements with weight {{nowrap|<math display="inline">w_i = 1 / {\sigma_i^2}</math>,}} and the resulting variance is smaller than each of the independent measurements {{nowrap|<math display="inline"> \sigma^2 = 1 / \sum_i w_i</math>.}} The [[maximum likelihood]] method weights the difference between fit and data using the same weights {{nowrap|<math>w_i</math>.}}

The [[expected value]] of a random variable is the weighted average of the possible values it might take on, with the weights being the respective [[probability|probabilities]]. More generally, the expected value of a function of a random variable is the probability-weighted average of the values the function takes on for each possible value of the random variable.

In [[linear regression|regressions]] in which the [[dependent variable]] is assumed to be affected by both current and lagged (past) values of the [[independent variable]], a [[distributed lag]] function is estimated, this function being a weighted average of the current and various lagged independent variable values. Similarly, a [[moving average model]] specifies an evolving variable as a weighted average of current and various lagged values of a random variable.