7.5. Two Variables

Let $X$ and $Y$ be two random variables and let $F_{(}X,Y)(x,y)$ be their joint CDF.

$$\begin{array}{r}\underset{\begin{array}{c}x\to -\mathrm{\infty }\\ y\to -\mathrm{\infty }\end{array}}{lim}{F}_{X,Y}(x,y)=0.\end{array}$$

$$\begin{array}{r}\underset{\begin{array}{c}x\to \mathrm{\infty }\\ y\to \mathrm{\infty }\end{array}}{lim}{F}_{X,Y}(x,y)=1.\end{array}$$

Right continuity:

$$\underset{x\to x_0^{+}}{lim}{F}_{X,Y}(x,y)=F_{X,Y}(x_0,y).$$

$$\underset{y\to y_0^{+}}{lim}{F}_{X,Y}(x,y)=F_{X,Y}(x,y_0).$$

The joint probability density function is given by $f_{X,Y}(x,y)$. It satisfies $f_{X,Y}(x,y)\ge 0$ and

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_{X,Y}(x,y)dydx=1.$$

The joint CDF and joint PDF are related by

$$F_{X,Y}(x,y)=\mathbb{P}(X\le x,Y\le y)={\int }_{-\mathrm{\infty }}^x{\int }_{-\mathrm{\infty }}^y{f}_{X,Y}(u,v)dvdu.$$

Further

$$\mathbb{P}(a\le X\le b,c\le Y\le d)={\int }_a^b{\int }_c^d{f}_{X,Y}(u,v)dvdu.$$

The marginal probability is

$$\mathbb{P}(a\le X\le b)=\mathbb{P}(a\le X\le b,-\mathrm{\infty }\le Y\le \mathrm{\infty })={\int }_a^b{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_{X,Y}(u,v)dvdu.$$

We define the marginal density functions as

$$f_X(x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_{X,Y}(x,y)dy$$

and

$$f_Y(y)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_{X,Y}(x,y)dx.$$

We can now write

$$\mathbb{P}(a\le X\le b)={\int }_a^b{f}_X(x)dx.$$

Similarly

$$\mathbb{P}(c\le Y\le d)={\int }_c^d{f}_Y(y)dy.$$

7.5.1. Conditional Density

We define

$$\mathbb{P}(a\le x\le b|y=c)={\int }_a^b{f}_{X|Y}(x|y=c)dx.$$

We have

$$f_{X|Y}(x|y=c)=\frac{f_{X,Y}(x,c)}{f_Y(c)}.$$

In other words

$$f_{X|Y}(x|y=c)f_Y(c)=f_{X,Y}(x,c).$$

In general we write

$$f_{X|Y}(x|y)f_Y(y)=f_{X,Y}(x,y).$$

Or even more loosely as

$$f(x|y)f(y)=f(x,y).$$

More identities

$$f(x|y\le d)=\frac{{\int }_{-\mathrm{\infty }}^{d}f(x,y)dy}{\mathbb{P}(y\le d)}.$$

7.5.2. Independent Variables

If $X$ and $Y$ are independent then

$$f_{X,Y}(x,y)=f_X(x)f_Y(y).$$

$$f(x|y)=\frac{f(x,y)}{f(y)}=\frac{f(x)f(y)}{f(y)}=f(x).$$

Similarly

$$f(y|x)=f(y).$$

The CDF also is separable

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