{\displaystyle \operatorname {E} [X]=\sum _{i=1}^{n}x_{i}\,p_{i}=x_{1}p_{1}+x_{2}p_{2}+\cdots +x_{n}p_{n}.}

where be a random variable with a finite number of finite outcomes occurring with probabilities respectively.

Since all probabilities add up to 1 (), the expected value is the weighted average, with ’s being the weights.

If all outcomes are equiprobable (that is, ), then the weighted average turns into the simple average.

See also