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.

Parents:

  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.