Expected Value
Expected-value problems are central to quant interviews because they test modeling and decomposition rather than arithmetic.
Lessons · recommended order
- 1Expected Value: FundamentalsThe weighted average of a random variable's outcomes, and how to compute it for discrete and continuous variables.Foundations8 min
- 2Linearity of ExpectationE[X+Y] = E[X] + E[Y] always holds — no independence required — making it the single most useful tool in quant interviews.Foundations8 min
- 3Indicator VariablesWrite a count as a sum of 0/1 indicators to turn combinatorial counting problems into simple probability calculations.Intermediate10 min
- 4Conditional ExpectationE[X|Y] is a random variable in Y, and averaging it over Y's distribution recovers E[X] — the law of total expectation.Intermediate9 min
- 5Tail-Sum FormulaFor a nonnegative integer-valued X, E[X] = Σ P(X ≥ k) — often far easier than summing k·P(X=k) directly.Intermediate8 min
- 6Recursive Expectation (First-Step Analysis)Set up an equation for E[X] by conditioning on the first step of a random process, then solve algebraically.Intermediate10 min
- 7Optimal StoppingCompare the value of stopping now against the expected value of continuing to decide the optimal threshold strategy.Advanced9 min
- 8The Coupon Collector ProblemExpected time to collect all n distinct coupons is nHₙ, derived by summing expected waiting times across phases.Intermediate8 min