For a **discrete** [[Random Variable|random variable]] $X$, the PMF is defined as a function mapping $ p_X(x) \equiv P(X=x) = P(\{\omega \in \Omega | X(\omega) = x\} \in \Sigma). $ This concept can be expanded to a vector of random variables, yielding the *[[Probability Distribution|joint probability distribution]]* $ p_{X_1,X_2,\dots,X_n}(x_1,x_2,\dots, x_n)=P(X_1=x_1, X_2=x_2,\dots ,X_n=x_n). $ Probability mass functions transform naturally along a [[Measurable Function|measurable function]] $g \colon X \rightarrow Y$ via $p(y)=\sum\limits_{x \in g^{-1}(y)}p_{X}(x)$