Marginal Distributions for Discrete Random Variables

Recall that a joint PMF $p_{X,Y}(x,y)=P(X=x,Y=y)$ specifies the probability of every pair $(x,y)$. To recover the distribution of just one of the variables, we sum out the other.

The marginal PMF of $X$ is

$$p_X(x)=P(X=x)=\sum_{y} p_{X,Y}(x,y),$$

where the sum runs over all values that $Y$ can take. This is the law of total probability applied to the partition $\{Y=y\}$. Similarly, the marginal PMF of $Y$ is

$$p_Y(y)=\sum_{x} p_{X,Y}(x,y).$$

When the joint PMF is displayed as a table, $p_X(x)$ is the sum of row $x$ and $p_Y(y)$ is the sum of column $y$ — the marginals appear literally in the margins of the table. For instance, with

$$\begin{array}{c|cc|c} & Y=0 & Y=1 & p_X(x) \\ \hline X=0 & 0.2 & 0.1 & 0.3 \\ X=1 & 0.4 & 0.3 & 0.7 \\ \hline p_Y(y) & 0.6 & 0.4 & 1 \end{array}$$

we read off $p_X(0)=0.2+0.1=0.3$ and $p_Y(1)=0.1+0.3=0.4$. Each marginal PMF must itself sum to $1$.