Marginal Probability Mass Function¶

Definition¶

For a discrete random pair $(X,Y)$ with joint PMF $p_{X,Y}$ , the marginal PMF of $X$ is

\begin{aligned} p_X(x)=\sum_y p_{X,Y}(x,y), \end{aligned}

where the sum runs over all possible values $y$ of $Y$ .

Interpretation¶

A marginal PMF describes one component of a random pair after the other component has been summed out. Summing over all values $y$ is the law of total probability for the partition formed by the events $\{Y=y\}$ $y}$ : $\displaystyle \mathbb{P}(X=x)=\sum_y\mathbb{P}(X=x,Y=y)$ $y)$ .

A 2 by 2 joint probability table with its row sums 0.3 and 0.7 as the marginal PMF of X and its column sums 0.4 and 0.6 as the marginal PMF of Y.

In the table the rows are the values of $X$ and the columns the values of $Y$ . The row sums $0.1+0.2=0.3$ and $0.3+0.4=0.7$ give $p_X$ ; the column sums $0.1+0.3=0.4$ and $0.2+0.4=0.6$ give $p_Y$ . The name comes from writing these sums in the margins of the table.

Going from the joint PMF to the marginals loses information: the marginals do not determine the joint PMF. The pair $X=Y\sim\operatorname{Bern}(\tfrac{1}{2})$ and a pair of independent fair coins have the same marginals but different joint PMFs. Only for independent variables is the joint PMF the product $p_X(x)p_Y(y)$ ; the table above is not of this kind, since $p_X(0)p_Y(0)=0.3\cdot0.4=0.12\neq0.1$ $0.1$ .

The marginal of $Y$ is the denominator of the conditional PMF of $X$ given $Y=y$ $y$ . For pairs with a joint density, the marginal density is found in the same way, with the sum replaced by an integral.

Key Results¶

$\displaystyle p_X(x)=\sum_yp_{X,Y}(x,y)$ and $\displaystyle p_Y(y)=\sum_xp_{X,Y}(x,y)$ $x \sum p_{X, Y} (x, y)$ .
Marginalization preserves normalization: $\displaystyle \sum_xp_X(x)=1$ $1$ .

Examples¶

If the masses at $(0,0),(0,1),(1,0),(1,1)$ are $0.1,0.2,0.3,0.4$ , then $p_X(0)=0.3$ and $p_Y(1)=0.6$ $0.6$ .
For independent fair dice, summing the six masses $1/36$ in a row gives $p_X(i)=1/6$ $1/6$ .

Prerequisites¶

Joint Probability Mass Function

Sources¶

Nicky van Foreest · Probability Distributions · Week 1

Aliases: marginal PMF, marginal mass function Symbols: sum

Last updated: 20 October 2026 03:02