Linearity of Expectation in Quant Interviews
How linearity of expectation solves quant interview problems involving sums, indicators, expected counts, and dependent events.
Candidates who over-enumerate expected value problems or avoid indicator variables.
The core idea
Linearity of expectation says E[X + Y] = E[X] + E[Y]. The variables do not need to be independent. That makes it one of the cleanest tools for expected counts, sums, and games with many overlapping events.
Use indicators
An indicator variable is one if an event happens and zero otherwise. Its expectation is the probability of the event. If you can express a count as a sum of indicators, the expected count becomes a sum of probabilities.
Dependence is not a blocker
Candidates often avoid linearity because events overlap. Overlap matters for variance, but not for expectation. You can add expected values even when the underlying events are dependent.
Concrete example
If five cards are drawn, the expected number of aces is 5 x (4/52) = 5/13. The card indicators are dependent, but linearity still works. Enumeration of every hand is unnecessary.
When not to stop there
Linearity gives expectation, not distribution. If the interviewer asks for probability of at least one event, variance, or risk, you need more than the expected count. Say what linearity answers and what it does not.
Common mistakes
Candidates either forget linearity and enumerate too much, or use expected value as if it were a probability guarantee. Keep the target clear: expectation is an average, not a promise about one trial.
Practice the pattern
Use the LeetQuidity curriculum and calibration to turn this topic into a focused practice plan.