TL;DR

The property that \( E[aX + bY] = aE[X] + bE[Y] \) regardless of whether X and Y are dependent. Extends to any finite or countably infinite sum of random variables.

By Valenke Exam Prep Team·Last updated 2026-06-03

Linearity of Expectation

The property that \( E[aX + bY] = aE[X] + bE[Y] \) regardless of whether X and Y are dependent. Extends to any finite or countably infinite sum of random variables.

Why it matters for interviews

The single most powerful tool for solving expected value problems in quant interviews. Combined with indicator random variables, it converts complex counting problems into simple sums.

Definition and Mathematical Foundation

The property that \( E[aX + bY] = aE[X] + bE[Y] \) regardless of whether X and Y are dependent. Extends to any finite or countably infinite sum of random variables.

Application in Quantitative Finance

The single most powerful tool for solving expected value problems in quant interviews. Combined with indicator random variables, it converts complex counting problems into simple sums.

Related Terms

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Frequently Asked Questions

Why is linearity of expectation so powerful?▼

It holds without independence, so you can decompose complex expectations into sums of simple ones without worrying about dependence structure. Many problems that seem hard become trivial with this technique.

How does the indicator variable trick work?▼

Express a count as \( X = \sum I_i \) where \( I_i \) are indicator variables. Then \( E[X] = \sum P(I_i = 1) \). For example, expected number of fixed points in a random permutation: \( \sum_{i=1}^n 1/n = 1 \).

Does linearity hold for variance?▼

No. \( \text{Var}(X+Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X,Y) \). Linearity of variance holds only for uncorrelated (not just independent) random variables.