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 $x<<(x+dx)$ 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.