Expected Value

E[X] = _x x · P(X = x) — value-weighted average.

The idea

The expected value of a random variable is the average of its possible values, weighted by their probabilities. It is the number that the running average of repeated, independent runs of the experiment settles toward.

An ordinary average counts every entry once, but the values of a random variable need not be equally likely, and a value that occurs nine times in ten should count for more than one that occurs once in ten. Weighting each value by its probability and adding gives $\mathbb{E}[X] = \sum_{x} x \cdot \mathbb{P}(X = x),$ read as the expectation of $X$. The weights sum to $1$, which makes the result an average rather than a total, and when the values are equally likely it reduces to the familiar mean.

$\mathbb{E}[X]$ need not be a value that $X$ can take: a coin scored $1$ for heads and $0$ for tails has expectation $\tfrac{1}{2}$. It is a balance point, not a prediction, and it says nothing about how widely the values spread around it.

Ways to work on it

• Walkthrough. Fair 4-sided die — three small steps.
• Practice. One step. Three-value weighted random variable.
• Hardest. One step. Four-value random variable — more terms to sum.

Not sure where to start? Take the ten-question placement test.