# conditional expectation

(1.8 hours to learn)

## Summary

The conditional expectation E[X | Y] is the expectation of X in the conditional distribution P(X | Y).

## Context

This concept has the prerequisites:

- expectation and variance
- conditional distributions (Iterated expectation involves taking expectations under conditional distributions.)

## Goals

- Know the definitions of conditional expectation for both discrete and continuous random variables

- Know and be able to apply the law of iterated expectations

## Core resources (read/watch one of the following)

## -Paid-

→ Mathematical Statistics and Data Analysis

An undergraduate statistics textbook.

Location:
Section 4.4, "Conditional expectation," pages 147-154

→ Probability and Statistics

An introductory textbook on probability theory and statistics.

Location:
Section 4.7, "Conditional expectation," pages 222-228

→ A First Course in Probability

An introductory probability textbook.

Location:
Section 7.5, "Conditional expectation," pages 365-382

## See also

- Some uses of iterated expectations:
- they justify the use of Monte Carlo estimators
- they are used in the definition of martingales