Mathematical Expectation

One of the most important characteristics of a random variable is its expectation — synonyms include expected value, mean, and first moment.

The idea is a direct generalisation of the ordinary numerical average. The average of $n$ numbers $a_1,a_2,\dots,a_n$ is $\dfrac{a_1+a_2+\cdots+a_n}{n}$ — a single value summarising the whole collection. If those $n$ numbers are re-viewed as the values of a random variable $X$ (each occurring with the frequency-based probability of $\tfrac1n$ per occurrence, or the appropriate relative frequency when values repeat), the same average can be rewritten as a probability-weighted sum: $\sum (\text{value})\times(\text{its probability})$. That reweighted sum — not $\tfrac1n$ per item, but the true probability of each value — is exactly the definition of $E(X)$ developed next; for a continuous random variable, the same weighting idea survives with the sum replaced by an integral. …