Cumulative Distribution Function

If $(X,Y)$ is a two dimensional random variable (discrete or continuous), then $F(x,y)=p\{X\le x\,andY\le y\}$ is called the cumulative distribution function of $(X,Y)$.

For discrete case,
$$F(x,y)=\sum\limits_{j}{\sum\limits_{i}{{{P}_{ij}}}}$$
$${{y}_{i}}\le y,{{x}_{i}}\le x$$

For continuous case,
$$F(x,y)=\int\limits_{-\infty }^{y}{\int\limits_{-\infty }^{x}{f(x,y)dxdy}}$$