Expected Value

Compact study note.

Summary

Expected value is the probability-weighted average of a random variable when the required sum or integral is finite. It is linear even when variables are dependent.^[1]

Prerequisites

Random Variable

Notation and Assumptions

Discrete expectation:

E [X] = \sum_{x} x P (X = x) .

Continuous expectation with density $f_{X}$ :

E [X] = \int_{- \infty}^{\infty} x f_{X} (x) d x .

Use these formulas when the absolute sum or integral is finite.

Essential Result

Linearity: $E [a X + b Y + c] = a E [X] + b E [Y] + c$ whenever the expectations exist.

Small Example

If $X$ counts heads in two fair flips, $E [X] = 0 (1 / 4) + 1 (1 / 2) + 2 (1 / 4) = 1$ .

Common Mistakes

Averaging possible values without probability weights.
Assuming expectation exists for every distribution.

Connections

Variance
Moment Generating Functions
Distributions

References

OpenStax, Introductory Statistics 2e, "Chapter 4: Discrete Random Variables", https://openstax.org/books/introductory-statistics-2e/pages/4-introduction ↩︎