Expected Value
Reference for expected value E[X] = Σ x · P(x).
The probability-weighted average of outcomes.
Foundation of decision theory, gambling, and insurance.
Discrete Random Variables
The expected value of a discrete random variable is the sum, over all possible values, of each value multiplied by its probability. It represents the long-run average if the random experiment were repeated many times.
Continuous Random Variables
For continuous variables with probability density function f(x), the sum becomes an integral over the support of X.
Variables
| Symbol | Meaning |
|---|---|
| X | A random variable |
| xᵢ | The i-th possible value of X |
| P(X = xᵢ) | Probability of outcome xᵢ |
| E[X] | Expected value of X |
| f(x) | Probability density function (continuous case) |
Example — Single Die Roll
Find the expected value of a single fair six-sided die.
Each face has probability 1/6
E[X] = 1 × 1/6 + 2 × 1/6 + 3 × 1/6 + 4 × 1/6 + 5 × 1/6 + 6 × 1/6
= (1 + 2 + 3 + 4 + 5 + 6) / 6 = 21 / 6
E[X] = 3.5
Note that 3.5 is not a value the die can actually show. Expected value is an average; it does not have to equal any individual outcome.
Example — Lottery Ticket
A lottery ticket costs $2. The jackpot is $10 million, with a 1 in 14 million chance of winning. Secondary prizes total an expected $0.30. Is the ticket worth buying based on expected value?
E[Jackpot] = $10,000,000 × (1 / 14,000,000) = $0.714
E[Total winnings] = $0.714 + $0.30 = $1.014
Net expected value = $1.014 − $2.00
Net E[X] = −$0.99 per ticket
On average, you lose 99 cents per ticket. Most state lotteries have negative expected value by design — that gap is the operator's profit margin.
Example — Insurance Pricing
An insurance company offers a $200,000 home fire policy. Historical data shows fires occur in 0.3% of policies per year, with average damage of $50,000. What is the actuarial fair premium (expected payout)?
P(fire) = 0.003 per year
E[payout] = 0.003 × $50,000 = $150 per policy per year
Fair premium ≈ $150 (before loading)
Actual premiums are higher to cover operating costs, claim handling, and profit. The "load" is typically 30-60% of the actuarial fair premium.
Properties of Expected Value
| Property | Statement |
|---|---|
| Linearity | E[aX + bY] = a × E[X] + b × E[Y] |
| Constant | E[c] = c for any constant c |
| Independence | If X and Y are independent: E[XY] = E[X] × E[Y] |
| Indicator | E[1_A] = P(A) |
| Monotonicity | If X ≤ Y always: E[X] ≤ E[Y] |
Linearity is the most useful property: expected value of a sum is the sum of expected values, even when the variables are not independent.
Example — Sum of Two Dice
Find the expected sum of two dice.
E[X + Y] = E[X] + E[Y] by linearity
= 3.5 + 3.5
E[X + Y] = 7
The most likely sum is also 7, but only because the distribution is symmetric — expected value and mode are not always equal.
When Expected Value Is Misleading
Expected value collapses an entire probability distribution into a single number. It does not capture risk, skewness, or extreme outcomes. Consider two bets:
- Bet A: 50% chance of +$10, 50% chance of −$10. E[X] = 0.
- Bet B: 1% chance of +$1,000,000, 99% chance of −$10,000. E[X] = $10,000 − $9,900 = $100.
Bet B has higher expected value, but bankrupts most players who attempt it. Expected value alone is the wrong decision rule in the presence of ruin risk. Variance, utility theory, and risk metrics (VaR, CVaR) extend the framework.
When to Use It
- Long-run average of a game of chance (gambling, lotteries, casino games)
- Actuarial fair pricing of insurance contracts
- Investment portfolio expected return calculations
- Decision making under uncertainty with repeatable choices
- Optimal stopping problems (when to sell, when to exercise an option)
- Algorithm analysis (expected running time, expected memory use)
Law of Large Numbers
The law of large numbers states that for a sequence of independent draws from the same distribution, the sample mean converges to the expected value as the number of draws grows. This is why expected value is meaningful: over many trials, your average outcome will approach E[X]. Over a single trial, anything can happen.