We define the joint cdf to be and conditional cdf to be Hence we get the following rules:

• Conditional probability (cdf):
  Fx|y=PrX≤x|Y≤y=FxyFYy F | x y Y y X x F x y FY y (3)
• Bayes Rule (cdf):
  Fx|y=Fy|xFxFy F | x y F | y x F x F y (4)
• Total probability (cdf):
  Fx∞=Fx F x F x (5)
  which follows because the event Y≤∞ Y itself forms a partition of the sample space.

Conditional cdf's have similar properties to standard cdf's, i.e. F X | Y -∞|y=0 F X | Y | y 0 F X | Y ∞|y=1 F X | Y | y 1

We define joint and conditional pdfs in terms of corresponding cdfs. The joint pad is defined to be and the conditional pdf is defined to be where F′x|Y=y=PrX≤x|Y=y F | x Y y Y y X x Note that F′x|Y=y F | x Y y is different from the conditional cdf Fx|Y=y F | x Y y , previously defined, but there is a slight problem. The event, Y=y Y y , has zero probability for continuous random variables, hence probability conditional on Y=y Y y is not directly defined and F′x|Y=y F | x Y y cannot be found by direct application of event-based probability. However all is OK if we consider it as a limiting case: Joint and conditional pdfs have similar properties and interpretation to ordinary pdfs: fxy>0 f x y 0 ∫∫fxydxdy=1 y x f x y 1 fx|y>0 f | x y 0 ∫fx|ydx=1 x f | x y 1 For pdfs we get the following rules:

• Conditional pdf:
  fx|y=fxyfy f | x y f x y f y (9)
• Bayes Rule (pdf):
  fx|y=fy|xfxfy f | x y f | y x f x f y (10)
• Total Probability (pdf):
  ∫fy|xfxdx=∫fyxdx=fy∫fx|ydx=fy x f | y x f x x f y x f y x f | x y f y (11)
  The final result is often referred to as the Marginalisation Integral and fy f y as the Marginal Probability.