A probability distribution describes uncertainty over possible numerical values. The expected value summarizes the distribution's center as a probability-weighted average.

For a discrete random variable $X$,

$$E[X]=\sum_x x,P(X=x),$$

when the sum is well defined.

Example

For a fair six-sided die,

$$E[X]=\frac{1+2+3+4+5+6}{6}=3.5.$$

The value $3.5$ can never appear on a single roll. Expectation is not a prediction of one outcome; it describes the average value approached over many repetitions of the same probability model.

Functions of a random variable

More generally, for a discrete variable,

$$E[g(X)]=\sum_x g(x)P(X=x).$$

This lets expectations describe quantities derived from the original random variable.

Linearity of expectation

Expectation behaves linearly:

$$E[aX+b]=aE[X]+b.$$

More generally,

$$E[X+Y]=E[X]+E[Y]$$

whenever the expectations exist. Independence is not required for this identity.

For continuous random variables, the same probability-weighted-average idea is expressed using a probability density and integration once those concepts are available.

Expectation is a property of the whole probability distribution, not a value that one individual observation is required to take.