Expected Value

The definition

For a discrete random variable X taking values x₁, x₂, ... with probabilities p₁, p₂, ...:

E[X] = ∑ xᵢ · pᵢ

For a continuous random variable with probability density f(x):

E[X] = ∫ x · f(x) dx

Either way, you weight each possible value by its probability and sum (or integrate). The result is the long-run average over many independent realizations of X.

Worked examples

Example 1 — fair six-sided die

X is the result of one roll. Possible values 1, 2, 3, 4, 5, 6 each with probability 1/6.

E[X] = (1+2+3+4+5+6)/6 = 21/6 = 3.5

Average over many rolls is 3.5. Notice — 3.5 is not even possible to roll. "Expected" is the long-run mean, not the typical single outcome.

Example 2 — coin flip with $1 win/$1 loss

P(heads) = 1/2, win $1; P(tails) = 1/2, lose $1.

E[X] = (1)(0.5) + (-1)(0.5) = 0

Fair game — average gain is zero. Over many flips, you break even (with variance — sometimes ahead, sometimes behind).

Example 3 — biased coin

P(heads) = 0.6, win $2; P(tails) = 0.4, lose $1.

E[X] = (2)(0.6) + (-1)(0.4) = 1.2 - 0.4 = $0.80 per flip

Positive expected value — over many flips, you gain $0.80 on average per flip. After 1000 flips, expected total winnings is $800.

Example 4 — lottery

$2 ticket. Jackpot $10 million with probability 1 in 100 million. (Smaller prizes ignored for simplicity.)

E[winnings] = 10,000,000 · 0.00000001 = $0.10
E[net] = $0.10 - $2 = -$1.90 per ticket

Each ticket loses $1.90 on average. The lottery is a tax on people bad at math (Voltaire's term, paraphrased). The expected dollar value is negative; the expected utility may be positive (the small dream of winning is itself worth something to many players).

Linearity of expectation

For any random variables X and Y (independent or not):

E[aX + bY] = a · E[X] + b · E[Y]

This is one of the most powerful identities in probability. It works without independence — even strongly correlated X and Y have E[X + Y] = E[X] + E[Y]. The variance, by contrast, requires independence — Var(X + Y) = Var(X) + Var(Y) only if X and Y are independent.

Linearity is the backbone of combinatorial probability. Want the expected number of fixed points in a random permutation of n elements? Define indicator variables I_k = 1 if k is fixed, 0 otherwise. E[I_k] = 1/n. Sum over k — E[number of fixed points] = n · 1/n = 1. Linearity gives this in one line; computing the actual distribution would take pages.

Variance and standard deviation

Expected value tells you the long-run average; variance measures the spread around it:

Var(X) = E[(X − E[X])²]
       = E[X²] − E[X]²       (computational form)

Standard deviation σ = √Var(X) is in the same units as X. For the fair die, Var = E[X²] − 3.5² = (1²+2²+...+6²)/6 − 12.25 = 91/6 − 12.25 ≈ 2.917; σ ≈ 1.71.

About 68% of values fall within 1σ of the mean for a normal distribution; broader for non-normal. The "spread" Var captures the typical deviation from the average.

Conditional expectation

The expected value of X given that some event A occurred:

E[X | A] = ∑ x · P(X = x | A)

For continuous variables, integrate against the conditional density. The Law of Total Expectation:

E[X] = E[E[X | Y]]

Compute the expected value of X conditional on Y, then take the expected value of that over Y. Useful when conditioning makes calculation easier — split a hard problem into cases based on Y, compute conditional expectation in each case, then average.

JavaScript — computing expected values

// Discrete expected value
function expectedDiscrete(values, probabilities) {
  let sum = 0;
  for (let i = 0; i < values.length; i++) {
    sum += values[i] * probabilities[i];
  }
  return sum;
}

// Fair die
const die = expectedDiscrete([1,2,3,4,5,6], Array(6).fill(1/6));
console.log(die);  // 3.5

// Lottery example
const lotteryNet = expectedDiscrete([10_000_000, 0], [0.00000001, 0.99999999]) - 2;
console.log(lotteryNet);  // -1.9

// Empirical estimate from samples
function empiricalMean(samples) {
  return samples.reduce((s, x) => s + x, 0) / samples.length;
}

// Simulate 1 million die rolls; converges to 3.5
const samples = Array.from({length: 1_000_000}, () => Math.floor(Math.random() * 6) + 1);
console.log(empiricalMean(samples));  // ≈ 3.500 (Law of Large Numbers)

Where expected value shows up

• Insurance. Premiums are set so E[claims] is less than E[premiums], with margin for operating costs and profit. Without expected-value math, insurance companies would go broke.
• Gambling and games. Casinos design games with negative expected value for the player. The "house edge" is exactly E[house gain per dollar wagered]. Some games (poker, sports betting) can have positive expected value for skilled players.
• Finance. Expected return of an investment, expected loss of a portfolio. Modern portfolio theory (Markowitz) optimizes expected return for given risk (variance).
• Reinforcement learning. The action-value function Q(s, a) is the expected discounted reward starting from state s with action a. The whole RL framework is built on optimizing expected returns.
• Algorithm analysis. Average-case time complexity is the expected number of operations over random inputs. Quicksort's "expected O(n log n)" is a statement about expected operation count.
• Decision theory. Pick the action with highest expected utility. Foundation of microeconomics, game theory, and rational choice models.
• Statistics. Maximum likelihood, Bayesian inference, hypothesis testing — all involve computing expectations over data distributions.

Common mistakes

• Treating expected value as the most likely value. They're different. The mean of a die is 3.5, which is impossible to roll. The mean of a heavy-tailed distribution can be extremely far from the median or mode.
• Confusing expected value with utility. $1M with certainty vs $0 or $2.5M with 50% each — same expected dollar value, but most people prefer the certain $1M (risk aversion). Expected utility theory captures this; expected value alone doesn't.
• Assuming linearity for variance. E[X+Y] = E[X] + E[Y] always. Var(X+Y) = Var(X) + Var(Y) ONLY when X and Y are independent. Forgetting this gives wrong variance estimates for correlated variables.
• Not accounting for finite samples. Empirical mean from finite samples differs from true expected value. Sample size needs to be large enough; standard error decreases as 1/√n.
• Ignoring infinite expected values. Some distributions (Cauchy, t-distribution with 1 degree of freedom) have undefined or infinite expected value. The integral defining E[X] doesn't converge. Means from such samples don't converge with sample size.
• Forgetting the Law of Large Numbers requires independence. Sample mean → true mean only if samples are i.i.d. (or at least uncorrelated). Highly autocorrelated samples (like time-series) converge much slower or not at all.