# Definition:Marginal Probability Mass Function

## Definition

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space.

Let $X: \Pr \to \R$ and $Y: \Pr \to \R$ both be discrete random variables on $\left({\Omega, \Sigma, \Pr}\right)$.

Let $p_{X, Y}$ be the joint probability mass function of $X$ and $Y$.

Then the probability mass functions $p_X$ and $p_Y$ are called the marginal (probability) mass functions of $X$ and $Y$ respectively.

The marginal mass function can be obtained from the joint mass function:

 $\ds p_X \left({x}\right)$ $=$ $\ds \Pr \left({X = x}\right)$ $\ds$ $=$ $\ds \sum_{y \mathop \in \operatorname{Im} \left({Y}\right)} \Pr \left({X = x, Y = y}\right)$ $\ds$ $=$ $\ds \sum_y p_{X, Y} \left({x, y}\right)$