Expected Value of a Discrete Random Variable

Expected value of a discrete random variable

The expected value E[X] of a discrete random variable is what you get when you weight every possible outcome by its probability and add them up: E[X] = Σ xᵢ · P(xᵢ). Think of it as the average you'd settle into if you ran the experiment over and over. One useful property is linearity, E[aX + b] = a·E[X] + b, and E[X + Y] = E[X] + E[Y] holds even when X and Y are dependent. A fair six-sided die, for instance, has E[X] = (1+2+3+4+5+6)/6 = 3.5. The St. Petersburg paradox is the famous edge case: E can run off to infinity, yet no sane person pays infinity to play. Marginal utility theory is what reconciles the two.

Applications

It turns up wherever uncertainty has to be priced: gambling EV analysis (the house edge), insurance premiums (the insurer comes out ahead on average), decisions made under risk, financial modeling of expected return, reinforcement learning that chases reward, actuarial work, portfolio theory, and A/B test design where you estimate the expected lift. Lotteries are the cautionary case, since E usually comes out negative and the average player loses money.

FAQ

Is the expected value always an attainable outcome? No. A die averages 3.5, yet no face actually reads 3.5. E[X] summarizes the distribution; it isn't necessarily a value you can roll.

Why do people play the lottery if E < 0? Risk preferences and utility theory explain it. A cheap ticket buys a tiny shot at a life-changing prize, and there's real value in spending a few days dreaming about it.

Must the probabilities sum to 1? For a valid distribution, yes. If yours don't, normalize them first by dividing each P(xᵢ) by ΣP(xᵢ).