Jointly Distributed Random Variables

Joint Probability Mass Function for Two Discrete Random Variables

• Let X and Y be two discrete random variables defined on a sample space S of an experiment. The joint p.m.f. p(x,y) is defined for each pair of numbers (x, y) by

p(x,y) = P(X=x and Y=y)

Let A be any set consisting of pairs of (x,y) values. Then the probability P[(X,Y)εA] is obtained by summing the joint pmf over pairs in A: ),(]).[( yxpAYXP

The marginal probability mass function

• MPMF of X and of Y, denoted by and respectively are given by

p

)(xpX )(ypY

y

X yxpxp ),()( x

Y yxpyp ),()(

Joint Probability density function

• Let X and Y be two continuous random variables Then the joint p.d.f. f(x,y) for X and Y if for any two dimensional set A

• In particular if A is the two dimensional rectangle

Then

A

dxdyyxfAYXP ),(]).[(

},:),{( dycbxayx

b

a

d

c

dxdyyxfdYcbXaPAYXP ),(),(],[(

Marginal PDF

• The marginal PDF of X and Y

yfordxyxfyf

xfordyyxfxf

Y

X

),()(

),()(

Expected values

• Let X and Y be jointly distributed rv’s with pmf p(x,y) or pdf f(x,y) according to whether the variables are discrete or continuous . Then the expected values of a function h(X,Y), denoted by E(h(X,Y)) is given by

E(h(X,Y))=

continuous are Y and X if ),(),(

discrete. are Y and X if ),(),(

dxdyyxfyxh

yxpyxhx y

Covariance

• The covariance between two rv’s X and Y is

.continuous YX, ,))(-(x

discrete YX, ),())((

))([(),(

X dxdyyxfy

yxpyx

YXEYXCov

Y

YX

YX

Correlation

• The correlation coefficient of X and Y denoted by Corr (X,Y), is defined by

XY

YXXY

YXCov

),(