Joint CDF

• Joint CDF can also be defined:

$$\begin{align} F_{X,Y}(x,y) &= \sum_{y'\leq y}\sum_{x'\leq x}p_{X,Y}(x',y')\\ F_{X,Y}(x,y) &= \int_{-\infty}^y\int_{-\infty}^xf_{X,Y}(x',y')dx'dy' \end{align}$$

• For independent variables, these sums/integrals can be factored so that:

$$F_{X,Y}(x,y) = F_{X}(x)F_{Y}(y)$$

• We can also obtain the marginal CDFs by setting the other variables to infinity:

$$F_X(x) = F_{X,Y}(x,\infty)\\ F_Y(y) = F_{X,Y}(\infty,y)$$

• Finally the fundamental theorem of calculus yields:

$$f_{X,Y}(x,y) = \frac{\partial^2}{\partial x \partial y}F_{X,Y}(x,y)$$