Independence of Discrete Random Variables

Define what it means for two discrete random variables to be independent, check independence from a joint pmf table, and use independence to compute joint probabilities from marginal information.

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Tutorial

Introduction

Two discrete random variables $X$ and $Y$ are independent if their joint pmf factors as the product of their marginal pmfs:

P(X=x,\, Y=y) \;=\; P(X=x)\cdot P(Y=y)

for every pair $(x,y)$ in the support of $(X,Y).$

Intuitively, knowing the value of one variable gives no information about the other. We write $X\perp Y$ to denote that $X$ and $Y$ are independent.

This condition must hold at every cell of the joint pmf table -- if even a single entry fails to equal the product of the corresponding marginals, then $X$ and $Y$ are not independent.

To check independence from a joint pmf table:

Sum each row to obtain the marginal pmf of $X,$ and each column to obtain the marginal pmf of $Y.$
For each cell, verify that $P(X=x,Y=y)=P(X=x)\cdot P(Y=y).$
If every cell matches, $X\perp Y.$ As soon as one cell fails, $X$ and $Y$ are not independent.

For example, consider the joint pmf

\begin{array}{c|cc} & Y{=}0 & Y{=}1 \\ \hline X{=}0 & 0.12 & 0.18 \\ X{=}1 & 0.28 & 0.42 \end{array}

The marginals are $P(X{=}0)=0.30,$ $P(X{=}1)=0.70,$ $P(Y{=}0)=0.40,$ and $P(Y{=}1)=0.60.$ Every cell equals the product of the corresponding marginals (for instance, $0.30\cdot 0.40=0.12$ ), so $X$ and $Y$ are independent.