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# Joint and Conditional cdfs and pdfs

Module by: Nick Kingsbury. E-mail the author

Summary: This module introduces joint and conditional cdfs and pdfs

## Cumulative distribution functions

We define the joint cdf to be

Fxy=Pr(Xx)(Yy) F x y X x Y y
(1)
and conditional cdf to be
Fx|y=PrXx| Yy F | x y Y y X x
(2)
Hence we get the following rules:
• Conditional probability (cdf):
Fx|y=PrXx| Yy =Fxy FY y 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

## Probability density functions

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

fxy=2Fxyxy f x y x y F x y
(6)
and the conditional pdf is defined to be
fx|y=dFx|Y=ydx f | x y x F | x Y y
(7)
where Fx|Y=y=PrXx| Y=y F | x Y y Y y X x Note that Fx|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 Fx|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:
Fx|Y=y=limit   δy 0 PrXx| y<Yy+δy =limit   δy 0 Fxy+δyFxy FY y+δy FY y=Fxyy fY y F | x Y y δ y 0 y Y y δ y X x δ y 0 F x y δ y F x y FY y δ y FY y y F x y fY y
(8)
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

### Note:

From now on interpret as - - unless otherwise stated.
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=fyfx|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.

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##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

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##### What are tags?

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