Mathematical Expectation of a Random Variable

The mathematical expectation (or expected value) of a discrete random variable $X$, denoted $E(X)$, is defined as:

That is, each possible value is multiplied by its own probability, and the results are added up. $E(X)$ is a single number that summarises the centre of the whole probability distribution — it is the weighted average of all possible values, where the weights are the probabilities.

Why "expectation" and not just "average"

$E(X)$ is often called the long-run average: if the underlying random experiment were repeated a very large number of times and the resulting values of $X$ were averaged, that average would settle down close to $E(X)$. It does not need to be a value $X$ can actually take — for instance, the expected number of heads in two coin tosses works out to $1$, but the expected number of sixes when a die is rolled several times could easily come out as a fraction like $2.5$, which is not a value a single roll can show.

Properties of mathematical expectation

These properties make expectation easy to work with algebraically, without recomputing a sum from scratch every time:

1. Expectation of a constant: $E(a) = a$ for any constant $a$ — a quantity that never varies has itself as its own average.
2. Expectation of a constant times a variable: $E(aX) = a\,E(X)$.
3. Expectation of a linear function: $E(aX + b) = a\,E(X) + b$, combining the two properties above.
4. Expectation of a sum: $E(X + Y) = E(X) + E(Y)$, for any two random variables $X$ and $Y$ — this holds whether or not $X$ and $Y$ are independent, which makes it one of the most useful shortcuts in the whole subject. More generally, $E(aX + bY + c) = a\,E(X) + b\,E(Y) + c$. …