SL 4.7—Discrete random variables

A discrete random variable $X$ takes countable numerical values with probabilities $P(X=x)$ satisfying $0\le P(X=x)\le1$ and $\sum P(X=x)=1$. Its expected value is $E(X)=\sum xP(X=x)$.

Complete a missing probability by making the total equal 1, then multiply each outcome by its probability and add. In a gain model, positive values are winnings and negative values are losses; $E(X)=0$ describes a fair game in the long run.

Example

A game pays \4$with probability$0.2$and loses$$1$with probability$0.8$. Then$E(X)=4(0.2)-1(0.8)=0$, so it is fair by expected gain, although any individual play still wins$4 or loses $1.

Expected value need not be an attainable outcome and does not predict one play. At this SL objective, do not import variance transformations from the separate AHL random-variable content.