Joint, Marginal, and Conditional Distributions

Compact study note.

Summary

One joint distribution describes random variables together. Marginal distributions summarize one variable; conditional distributions describe one variable after another value or event is known.^[1]

Prerequisites

• Random Variable
• Conditional Probability

Notation and Assumptions

For discrete variables, use $p_{X,Y}(x,y)=P(X=x,Y=y)$ . Marginals are $p_X(x)=\sum _y{p}_{X,Y}(x,y)$ and $p_Y(y)=\sum _x{p}_{X,Y}(x,y)$ .

Essential Result

When $p_Y(y)>0$ , $p_{X\mid Y}(x\mid y)=p_{X,Y}(x,y)/p_Y(y)$ . Continuous versions replace sums by integrals and PMFs by densities.

Small Example

If $p_{X,Y}(0,0)=0.2$ , $p_{X,Y}(1,0)=0.3$ , $p_{X,Y}(0,1)=0.1$ , $p_{X,Y}(1,1)=0.4$ , then $p_X(1)=0.7$ .

Common Mistakes

• Confusing joint probability with conditional probability.
• For continuous variables, treating $f_{X,Y}(x,y)$ as point probability.

Connections

• Conditional Probability
• Independence
• Covariance and Correlation

References