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Joint and Conditional cdfs and pdfs

Module by: Nick Kingsbury

Summary: This module introduces joint and conditional cdfs and pdfs

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Cumulative distribution functions

We define the joint cdf to be

Fxy=PrX≤x∧Y≤y F x y X x Y y (1)
and conditional cdf to be
Fx|y=PrX≤x|Y≤y F | x y Y y X x (2)
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

Probability density functions

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

fxy=∂2∂x∂yFxy f x y x y F x y (6)
and the conditional pdf is defined to be
fx|y=∂∂xF′x|Y=y f | x y x F | x Y y (7)
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:
F′x|Y=y=limδy→0PrX≤x|y<Y≤y+δy=limδy→0Fxy+δy−FxyFYy+δy−FYy=∂∂yFxyfYy 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=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.

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