Joint and Conditional cdfs and pdfs 2.7 2002/12/20 2003/03/25 Nick Kingsbury ngk10@cam.ac.uk Liqun Wang liqun@rice.edu Nick Kingsbury ngk10@cam.ac.uk conditional cdf conditional pdf joint cdf joint pdf This module introduces joint and conditional cdfs and pdfs

Cumulative distribution functions We define the joint cdf to be F x y X x Y y and conditional cdf to be F | x y Y y X x Hence we get the following rules: Conditional probability (cdf): F | x y Y y X x F x y FY y Bayes Rule (cdf): F | x y 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. Conditional cdf's have similar properties to standard cdf's, i.e. F X | Y | y 0 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 f x y x y F x y and the conditional pdf is defined to be f | x y x F | x Y y where F | x Y y Y y X x Note that F | x Y y 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 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: 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 Joint and conditional pdfs have similar properties and interpretation to ordinary pdfs: f x y 0 y x f x y 1 f | x y 0 x f | x y 1 From now on interpret ∫ as ∫ - ∞ ∞ unless otherwise stated. For pdfs we get the following rules: Conditional pdf: f | x y f x y f y Bayes Rule (pdf): f | x y f | y x f x f y Total Probability (pdf): x f | y x f x x f y x f y x f | x y f y The final result is often referred to as the Marginalisation Integral and f y as the Marginal Probability.