1.3 Joint and conditional cdfs and pdfs

Cumulative distribution functions

We define the joint cdf to be

• 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) :
  $F(x)$∞ F x
  which follows because the event $Y$∞ itself forms a partition of the sample space.

Probability density functions

We define joint and conditional pdfs in terms of corresponding cdfs. The joint pad is defined to be

• 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 .