# Connexions

You are here: Home » Content » Jointly Distributed Random Variables

### Recently Viewed

This feature requires Javascript to be enabled.

# Jointly Distributed Random Variables

Module by: Don Johnson. E-mail the author

Two (or more) random variables can be defined over the same sample space. Just as with jointly defined events, the joint distribution function is easily defined.

P X , Y xyPrXxYy P X Y x y X x Y y
(1)
The joint probability density function p X Y xy p X Y x y is related to the distribution function via double integration.
P X , Y xy=xyp X Y αβdαdβ P X Y x y β x α y p X Y α β
(2)
or p X Y xy=2P X , Y xyxy p X Y x y x y P X Y x y Since limit  yP X , Y xy=P X x y P X Y x y P X x , the so-called marginal density functions can be related to the joint density function.
p X x=p X Y xβdβ p X x β p X Y x β
(3)
and p Y y=p X Y αydα p Y y α p X Y α y

Extending the ideas of conditional probabilities, the conditional probability density function p X | Y x | Y = y p X | Y x | Y = y is defined (when p Y y0 p Y y 0 ) as

p X | Y x | Y = y =p X Y xyp Y y p X | Y x | Y = y p X Y x y p Y y
(4)
Two random variables are statistically independent when p X | Y x | Y = y =p X x p X | Y x | Y = y p X x , which is equivalent to the condition that the joint density function is separable: p X Y xy=p X xp Y y p X Y x y p X x p Y y .

For jointly defined random variables, expected values are defined similarly as with single random variables. Probably the most important joint moment is the covariance:

covXYEXYEXEY cov X Y X Y X Y
(5)
where EXY=xyp X Y xydxdy X Y y x x y p X Y x y Related to the covariance is the (confusingly named) correlation coefficient: the covariance normalized by the standard deviations of the component random variables. p X , Y =covXY σ X σ Y p X , Y cov X Y σ X σ Y When two random variables are uncorrelated, their covariance and correlation coefficient equals zero so that EXY=EXEY X Y X Y . Statistically independent random variables are always uncorrelated, but uncorrelated random variables can be dependent. 1

A conditional expected value is the mean of the conditional density.

EX| Y = p X | Y x | Y = y dx Y X x p X | Y x | Y = y
(6)
Note that the conditional expected value is now a function of YY and is therefore a random variable. Consequently, it too has an expected value, which is easily evaluated to be the expected value of XX.
EEX| Y =x p X | Y x | Y = y dxp Y ydy=EX Y X y x x p X | Y x | Y = y p Y y X
(7)
More generally, the expected value of a function of two random variables can be shown to be the expected value of a conditional expected value: EfXY=EEfXY| Y f X Y Y f X Y . This kind of calculation is frequently simpler to evaluate than trying to find the expected value of fXY f X Y "all at once." A particularly interesting example of this simplicity is the random sum of random variables. Let LL be a random variable and X l X l a sequence of random variables. We will find occasion to consider the quantity l=1L X l l 1 L X l . Assuming that each component of the sequence has the same expected value EX X , the expected value of the sum is found to be
E S L =EEl=1L X l | L =ELEX=ELEX S L L l 1 L X l L X L X
(8)

## Footnotes

1. Let XX be uniformly distributed over -1 1 -1 1 and let Y=X2 Y X 2 . The two random variables are uncorrelated, but are clearly not independent.

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

### Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### 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.

##### Who can create a lens?

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

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks