Skip to content Skip to navigation

Connexions

You are here: Home » Content » Joint and Conditional cdfs and pdfs

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

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

      What is in a lens?

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

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

    • External bookmarks
  • E-mail the author

Recently Viewed

Joint and Conditional cdfs and pdfs

Module by: Nick Kingsbury

Summary: This module introduces joint and conditional cdfs and pdfs

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.

Comments, questions, feedback, criticisms?

Send feedback