Expectation (Mean) of a Random Variable

The expectation or mean of a discrete random variable $X$, denoted $E(X)$ or $\mu$, is the probability-weighted average of the values $X$ can take. It tells us the long-run average outcome if the random experiment were repeated a very large number of times.

Properties of expectation (useful for shortcuts):

• $E(c) = c$ for any constant $c$.
• $E(aX) = a\,E(X)$ for any constant $a$.
• $E(aX + b) = a\,E(X) + b$ for constants $a, b$ — a linear change of scale/origin in $X$ produces the same linear change in its mean.

Example. For the two-coin experiment of the previous section ($X = 0,1,2$ with probabilities $\tfrac14, \tfrac12, \tfrac14$): …