5.2 Cumulative Distribution Function

The joint (cumulative) distribution function of $X$ and $Y$ is defined as

i.e. the probability that a random variable $X$ takes a value less than or equal to $x$ and $Y$ takes a value less than or equal to $y$.

Unnumbered Figure: Link

Properties of $F_{XY}(x,y)$:

1. 1.

   This is defined for all random variables, i.e. discrete, continuous or a mixture of these.

2. 2.

   Since it is a probability: $0\le F_{XY}(x,y)\le 1$ for all $x$ and $y$, and

   1. $F_{XY}(-\mathrm{\infty },y)=0$,

   2. $F_{XY}(x,-\mathrm{\infty })=0$,

   3. $F_{XY}(\mathrm{\infty },\mathrm{\infty })=1$.

3. 3.

   Quiz: $F_{XY}(x,\mathrm{\infty })=[0$ or $F_X\mathit{}\mathrm{(}x\mathrm{)}$ or $\mathrm{1}\mathrm{]}$? $F_X(x)$ Similarly $F_{XY}(\mathrm{\infty },y)=F_Y(y)$

4. 4.

   $F_{XY}(x,y)$ is non-decreasing in both $x$ and $y$, i.e. for all $h\ge 0$

   1. $F_{XY}(x+h,y)\ge F_{XY}(x,y)$,

   2. $F_{XY}(x,y+h)\ge F_{XY}(x,y)$.

The probability of $(X,Y)$ falling in a rectangle with opposite corners $(x_1,y_1)$ and $(x_2,y_2)$, with and , can be found from $F_{XY}$ using

Unnumbered Figure: Link