> [!NOTE] Definition 1 (Joint Probability Mass Function of Bivariate Discrete Real-Valued Random Variables) > 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 $(\Omega, \mathcal{F}, \mathbb{P}).$ The *joint probability mass function* of $X$ and $Y$ is the [[Real-Valued Function on Real n-Space (Multivariable Function)|real-valued function]] $p_{X,Y}:\mathbb{R}^{2}\to \mathbb{R}$ defined by $p_{X,Y}(x,y)=\mathbb{P}(X=x, Y=y)$where $X=x,Y=y$ denotes the set $\{ \omega\in \Omega: (X(\omega) =x) \land (Y(\omega)=y) \}.$ **Note:** $X=x, Y=y$ denotes $X^{-1}(x) \cap Y^{-1}(y)$ which is an event since [[Event Spaces are Closed Under Finite Intersections|event spaces are closed under finite intersections]] and by definition $X^{-1}(x)$ and $Y^{-1}(y)$ are events. > [!NOTE] Definition (General) > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. Let $n\in \mathbb{N}^{+}$ and $X=\{X_{1},X_{2},\dots ,X_{n} \}$ be a set of [[Discrete random variables|discrete real-valued random variables]] on $(\Omega, \mathcal{F}, \mathbb{P}).$ The *joint probability mass function* of $X$ is the [[Real-Valued Function on Real n-Space (Multivariable Function)|real-valued function]] $p_{X}:\mathbb{R}^{n}\to \mathbb{R}$ where for all $x=(x_{1},x_{2},\dots,x_{n})\in \mathbb{R}^{n}$ $p_{X}(x)=\mathbb{P}(X_{1}=x_{1},X_{2}=x_{2},\dots, X_{n}=y_{n}).$ # Properties # Applications **Generalisation**: [[Joint Probability Mass Function]]. **Marginal probability function**: The PMFs $p_{X}$ and $p_{Y}$ are called the **marginal probability mass functions** of $X$ and $Y$ respectively. The [[Marginal Probability Mass Function of Discrete Real-Valued Random Variable|marginal mass functions]] can be obtained from the joint mass function: $p_{X}(x)=\sum_{y\in D_{Y}} p_{X,Y}(x,y).$ **Expectation**: By [[Expectation of Real-Valued Function of Bivariate Discrete Real-Valued Random Variable]], for all $g:\mathbb{R}^{2}\to \mathbb{R},$ $\mathbb{E}[g(X,Y)]=\sum_{x\in D_{X}}\sum_{y\in D_{Y}}g(x,y)\cdot p_{X,Y}(x,y).$ **Independent Variables**: Two discrete real-valued random variables $X$ and $Y$ are [[Independence of Two Discrete Real-Valued Random Variables|independent]] iff there Joint PMF satisfies $p_{X,Y}(x,y)=p_{X}(x)\cdot p_{Y}(y)$ for all $x,y\in \mathbb{R}.$