# conditional distributions

(1.5 hours to learn)

## Summary

The conditional distribution of a random variable X given another random variable Y is the distribution of X when Y is observed to take some vaule. While the precise mathematical definition is involved, for discrete and continuous variables, it amounts to dividing the joint PDF or PMF of X and Y by the PDF or PMF of Y.

## Context

This concept has the prerequisites:

## Goals

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

- For continuous random variables, why isn't it mathematically rigorous to condition on an event of probability zero?

- Know how the joint distribution of a set of random variables decomposes as a product of conditional distributions

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

## -Paid-

→ A First Course in Probability

An introductory probability textbook.

- Section 6.4, "Conditional distributions: discrete case," pages 288-291
- Section 6.5, "Conditional distributions: continuous case," pages 291-296

→ Mathematical Statistics and Data Analysis

An undergraduate statistics textbook.

Location:
Section 3.5, "Conditional distributions," pages 87-95

→ Probability and Statistics

An introductory textbook on probability theory and statistics.

Location:
Section 3.6, "Conditional distributions," pages 136-145

## See also

- We may be interested in whether two variables are independent conditioned on another random variable .