Expected value

Definition

The expectation of V (often shortened to "the expected V") is how much V you expect to get on average. For example, the expectation of a payoff, or an expected payoff, is how much money you will get on average; the expectation of the duration of a speech, or an expected duration, is how long the speech will last "on average."

Suppose V has discrete possible values, say $V=x_1,$ or $V=x_2,...,$ or $V=x_k$. Let $P\left(x_i\right)$ refer to the probability that $V=x_i$. Then the expectation of V is given by:

$$\sum _{i=1}^k{x}_{i}P\left(x_i\right)$$

Suppose V has continuous possible values, x. For instance, let $x\in R$. Let $P\left(x\right)$ be the continuous probability distribution, or ${lim}_{dx\to 0}$ of the probability that $x<V<(x+dx)$ divided by $dx$. Then the expectation of V is given by:

$${\int }_{-\infty }^{\infty }xP\left(x\right)dx$$

Importance

A common principle of reasoning under uncertainty is that if you are trying to achieve a good G, you should choose the act that maximizes the expectation of G.