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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

F⁢xy=Pr(X≤x)∧(Y≤y) F x y X x Y y
(1)
and conditional cdf to be
F⁢x|y=PrX≤x| Y≤y F | x y Y y X x
(2)
Hence we get the following rules:
  • Conditional probability (cdf):
    F⁢x|y=PrX≤x| Y≤y =F⁢xy FY ⁢y F | x y Y y X x F x y FY y
    (3)
  • Bayes Rule (cdf):
    F⁢x|y=F⁢y|x⁢F⁢xF⁢y F | x y F | y x F x F y
    (4)
  • Total probability (cdf):
    F⁢x∞=F⁢x 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

f⁢xy=∂2F⁢xy∂x∂y f x y x y F x y
(6)
and the conditional pdf is defined to be
f⁢x|y=∂dF⁢x|Y=yd∂x 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 F⁢x|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=limit   δ⁢y → 0 PrX≤x| y<Y≤y+δ⁢y =limit   δ⁢y → 0 F⁢xy+δ⁢y−F⁢xy FY ⁢y+δ⁢y− FY ⁢y=∂F⁢xy∂y 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: f⁢xy>0 f x y 0 ∫∫f⁢xydxdy=1 y x f x y 1 f⁢x|y>0 f | x y 0 ∫f⁢x|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:
    f⁢x|y=f⁢xyf⁢y f | x y f x y f y
    (9)
  • Bayes Rule (pdf):
    f⁢x|y=f⁢y|x⁢f⁢xf⁢y f | x y f | y x f x f y
    (10)
  • Total Probability (pdf):
    ∫f⁢y|x⁢f⁢xdx=∫f⁢yxdx=f⁢y⁢∫f⁢x|ydx=f⁢y 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 f⁢y f y as the Marginal Probability.

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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.

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