Expected value

The expected value of an action is the [-mean] numerical outcome of the possible results weighted by their Probability. It may actually be impossible to get the expected value, for example, if a coin toss decides between you getting $0 and $10, then we say you get "$5 in expectation" even though there is no way for you to get $5.

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 $P(x_{i})$ refer to the probability that $V = x_{i}$. Then the expectation of V is given by:

$$\sum_{i=1}^{k}x_{i}P(x_{i})$$

Suppose V has continuous possible values, x. For instance, let $x \in \mathbb{R}$. Let $P(x)$ 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:

$$\int_{-∞}^{∞}xP(x)dx$$

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.