Expected Value
Statistics & ProbabilityExpected value is the long-run average outcome of a random variable, calculated by weighting each possible value by its probability.
Formula
E(X) = \sum (\text{each value} \times \text{its probability})
Definition
Expected value is the average result you would get if you repeated an experiment many, many times. It is what you "expect" on average in the long run.
Example
A game: roll a die and win that many dollars. $E(\text{win}) = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 21/6 = \$3.50$. On average, you win $\$3.50$ per roll.
Key Insight
Expected value might not be a value that can actually occur. You cannot roll $3.5$ on a die, but $\$3.50$ is the long-run average.
Definition
The expected value $E(X) = \sum x_i P(X=x_i)$ for a discrete random variable. It is the probability-weighted average of all possible values. Expected value is used in decision theory, gambling, insurance, and economics to compare choices with uncertain outcomes.
Example
A lottery ticket costs $\$2$ and pays: $\$100$ with $P=0.01$, $\$10$ with $P=0.05$, $\$0$ with $P=0.94$. $E(\text{payout}) = 100(0.01) + 10(0.05) + 0(0.94) = 1 + 0.5 = \$1.50$. Expected profit $= \$1.50 - \$2.00 = -\$0.50$. On average, you lose $50$ cents per ticket.
Key Insight
If a game has a negative expected value, you lose money in the long run regardless of short-term luck. Casinos design all games with negative expected value for players.
Definition
The expected value $E[X] = \int x f(x)\,dx$ for continuous $X$, or $\sum x_i p_i$ for discrete $X$. Properties: linearity $E[aX+bY] = aE[X]+bE[Y]$, and for independent $X,Y$: $E[XY] = E[X]E[Y]$. In decision theory, expected utility maximization generalizes expected value by applying a utility function $u$: $E[u(X)]$ instead of $E[X]$.
Example
The St. Petersburg paradox: a game where you flip a coin until heads appears, winning $\$2^n$ if heads first appears on flip $n$. $E(\text{winnings}) = \sum_{n=1}^{\infty} (1/2^n)2^n = \sum 1 = \infty$. Despite infinite expected value, most people would not pay more than $\$20$-$\$30$ to play, illustrating the need for expected utility rather than expected value.
Key Insight
Jensen's inequality: for a concave utility function $u$ (risk aversion), $E[u(X)] \le u(E[X])$. Risk-averse decision-makers prefer a certain gain of $E[X]$ over the lottery $X$, even though both have the same expected value. This is the mathematical foundation of insurance markets.