## Definition A ==Joint Probability Function== describes the probability that two or more random variables simultaneously take specific values. For discrete random variables $X$ and $Y$, the joint PMF is $p(x, y) = P(X = x, Y = y)$. For continuous random variables, the joint PDF $f(x,y)$ satisfies: $ P((X,Y) \in A) = \iint_A f(x,y)\, dx\, dy $ > [!example]- Two dice are rolled. Let $X$ be the first die and $Y$ the second. What is $P(X = 3, Y = 5)$? > Since the dice are independent and each face has probability $1/6$: > $ P(X = 3, Y = 5) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36} $