> [!NOTE] Theorem (Marginal Probability Mass Function) > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. Let $X,Y$ be [[Discrete random variables|discrete real-valued random variables]] on $(\Omega, \mathcal{F}, \mathbb{P}).$ Let $p_{X,Y}$ denote the *[[Joint Probability Mass Function of Discrete Real-Valued Random Variables|joint probability mass function]]* of $X$ and $Y.$ Let $p_{X}$ denote the [[Probability Mass Function of Discrete Real-Valued Random Variable|probability mass function]] of $X.$ Then $p_{X}=\sum_{y\in D_{Y}}p_{X,Y}(x,y),$where $D_{Y}$ denotes the [[Discrete Support of Distribution of Discrete Real-Valued Random Variable|support]] of $p_{Y}$ and $\sum_{y\in D_{Y}}p_{X,Y}(x,y)$ is called the *marginal probability mass function* of $X$. **Proof**: Let $x\in \mathbb{R}.$ Then by [[Law of Total Probability]], $\begin{align} p_{X}(x) &= \mathbb{P}(X=x) \\ &= \sum_{y \in D_{Y}} \mathbb{P}(X=x,Y=y) + \mathbb{P}(X=x, Y \not \in D_{Y}) \\ &= \sum_{y \in D_{Y}} \mathbb{P}(X=x,Y=y) \\ &= \sum_{y \in D_{Y}} p_{X,Y}(x,y) \end{align}$