Joint Cumulative Distribution Functions

Definition

Let $X, Y: \Omega \to \R $ be random variables on probability space $(\Omega , \mathcal{F} , \mathbfsf{P} )$. The joint cumulative distribution function denoted $\rvcdf{X,Y}: \R ^2 \to [0, 1]$ is defined by $\rvcdf{X,Y}(s,t) = \mathbfsf{P} [X \leq s, Y \leq t]$.

In general the joint cdf for a random vector $X: \Omega \to \R ^n$ is $\rvcdf{X}: \R ^d \to [0,1]$ defined by $\rvcdf{X}(t) = \mathbfsf{P} [X \leq t]$ where $x \leq t$ means $x_i \leq t_i$ for all $i$.