This module introduces joint and conditional cdfs and pdfs
Cumulative distribution functions
We define the
joint cdf to be
$F(x, y)=((X\le x)\land (Y\le y))$
and
conditional cdf to be
$F((x, y))=(Y\le y, X\le x)$
Hence we get the following rules:

Conditional probability (cdf) :
$F((x, y))=(Y\le y, X\le x)=\frac{F(x, y)}{{F}_{Y}(y)}$

Bayes Rule (cdf) :
$F((x, y))=\frac{F((y, x))F(x)}{F(y)}$

Total probability (cdf) :
which follows because the event
$Y$∞ itself forms a partition of the sample space.
Conditional cdf's have similar properties to standard cdf's,
i.e.
$${F}_{XY}((()))$$∞
y
0
$${F}_{XY}(())$$∞
y
1
Probability density functions
We define joint and conditional pdfs in terms of corresponding
cdfs. The
joint pad is defined to be
$f(x, y)=\frac{\partial^{2}F(x, y)}{\partial x\partial y}$
and the
conditional pdf is defined to be
$f((x, y))=\frac{\partial^{1}\frac{d F((x, Y=y))}{d}}{\partial x}$
where
$$\frac{d F((x, Y=y))}{d}=(Y=y, X\le x)$$ Note that
$\frac{d F((x, Y=y))}{d}$ is different from the conditional cdf
$F((x, Y=y))$ , previously defined, but there is a slight
problem. The event,
$Y=y$ , has zero probability for continuous random
variables, hence probability conditional on
$Y=y$ is not directly defined and
$\frac{d F((x, Y=y))}{d}$ cannot be found by direct application of eventbased
probability. However all is OK if we consider it as a limitingcase:
$\frac{d F((x, Y=y))}{d}=\lim_{\delta (y)\to 0}(y< Y\le y+\delta (y), X\le x)=\lim_{\delta (y)\to 0}\frac{F(x, y+\delta (y))F(x, y)}{{F}_{Y}(y+\delta (y)){F}_{Y}(y)}=\frac{\frac{\partial^{1}F(x, y)}{\partial y}}{{f}_{Y}(y)}$
Joint and conditional pdfs have similar properties andinterpretation to ordinary pdfs:
$$f(x, y)> 0$$
$$\int \int f(x, y)\,d x\,d y=1$$
$$f((x, y))> 0$$
$$\int f((x, y))\,d x=1$$
From now on interpret
$\int $ as
${\int}_{\infty}^{\infty}$ unless otherwise stated.
For pdfs we get the following rules:

Conditional pdf:
$f((x, y))=\frac{f(x, y)}{f(y)}$

Bayes Rule (pdf):
$f((x, y))=\frac{f((y, x))f(x)}{f(y)}$

Total Probability (pdf):
$\int f((y, x))f(x)\,d x=\int f(y, x)\,d x=f(y)\int f((x, y))\,d x=f(y)$
The final result is often referred to as the
Marginalisation Integral and
$f(y)$ as the
Marginal Probability .