Expected value (E[X]) is the long-run average outcome of a random variable. It's calculated by summing each outcome multiplied by its probability.
Formula
E[X] = Σ(x_i × P(x_i))
Multiply each possible value by its probability, then sum all products. For continuous variables, integrate rather than sum.
All probabilities must sum to exactly 1 for a valid distribution.
Same expected value can have different risk profiles - consider variance too.
Positive expected value decisions lead to long-term success.
Compare expected returns of different investment options.
Determine optimal strategies in probabilistic scenarios.
Calculate premiums based on expected claim payouts.
Yes! For a fair die, E[X] = 3.5, which is impossible to roll. It represents the theoretical long-run average.
Negative expected value means you'll lose money long-term. Casinos always have positive expected value (for them).
Usually negative. A $1 ticket with 1/1,000,000 chance of $500,000 has expected value of -$0.50.
Var[X] = E[X²] - (E[X])². Variance measures spread around the expected value.