## Definition The ==Joint Cumulative Distribution Function== of random variables $X$ and $Y$ gives the probability that $X$ is at most $x$ and $Y$ is at most $y$ simultaneously: $ F(x, y) = P(X \leq x, Y \leq y) $ For continuous random variables, $F(x, y) = \int_{-\infty}^{x}\int_{-\infty}^{y} f(s, t)\, dt\, ds$, and the joint PDF can be recovered by $f(x,y) = \frac{\partial^2 F}{\partial x \,\partial y}$. > [!example]- If $F(x,y) = xy(x + y)$ for $0 \leq x \leq 1$ and $0 \leq y \leq 1$, what is $P(X \leq 0.5, Y \leq 0.5)$? > $ F(0.5, 0.5) = (0.5)(0.5)(0.5 + 0.5) = 0.25 \times 1 = 0.25 $