Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

110 Multivariate Distributions

For any fixedx 1 withpX 1 (x 1 )>0, this functionpX 2 |X 1 (x 2 |x 1 ) satisfies the con-

ditions of being a pmf of the discrete type becausepX 2 |X 1 (x 2 |x 1 ) is nonnegative

and

∑

x 2

pX 2 |X 1 (x 2 |x 1 )=

∑

x 2

pX 1 ,X 2 (x 1 ,x 2 )

pX 1 (x 1 )

=

1

pX 1 (x 1 )

∑

x 2

pX 1 ,X 2 (x 1 ,x 2 )=

pX 1 (x 1 )

pX 1 (x 1 )

=1.

We callpX 2 |X 1 (x 2 |x 1 )theconditional pmfof the discrete type of random variable
X 2 , given that the discrete type of random variableX 1 =x 1. In a similar manner,
providedx 2 ∈SX 2 , we define the symbolpX 1 |X 2 (x 1 |x 2 )bytherelation

pX 1 |X 2 (x 1 |x 2 )=

pX 1 ,X 2 (x 1 ,x 2 )

pX 2 (x 2 )

,x 1 ∈SX 1 ,

and we callpX 1 |X 2 (x 1 |x 2 ) the conditional pmf of the discrete type of random variable
X 1 , given that the discrete type of random variableX 2 =x 2. We often abbreviate
pX 1 |X 2 (x 1 |x 2 )byp 1 | 2 (x 1 |x 2 )andpX 2 |X 1 (x 2 |x 1 )byp 2 | 1 (x 2 |x 1 ). Similarly,p 1 (x 1 )
andp 2 (x 2 ) are used to denote the respective marginal pmfs.
Now letX 1 andX 2 denote random variables of the continuous type and have the
joint pdffX 1 ,X 2 (x 1 ,x 2 ) and the marginal probability density functionsfX 1 (x 1 )and
fX 2 (x 2 ), respectively. We use the results of the preceding paragraph to motivate
a definition of a conditional pdf of a continuous type of random variable. When
fX 1 (x 1 )>0, we define the symbolfX 2 |X 1 (x 2 |x 1 )bytherelation

fX 2 |X 1 (x 2 |x 1 )=

fX 1 ,X 2 (x 1 ,x 2 )

fX 1 (x 1 )

. (2.3.2)

In this relation,x 1 is to be thought of as having a fixed (but any fixed) value for

whichfX 1 (x 1 )>0. It is evident thatfX 2 |X 1 (x 2 |x 1 ) is nonnegative and that

∫∞

−∞

fX 2 |X 1 (x 2 |x 1 )dx 2 =

∫∞

−∞

fX 1 ,X 2 (x 1 ,x 2 )

fX 1 (x 1 )

dx 2

=

1

fX 1 (x 1 )

∫∞

−∞

fX 1 ,X 2 (x 1 ,x 2 )dx 2

=

1

fX 1 (x 1 )

fX 1 (x 1 )=1.

That is,fX 2 |X 1 (x 2 |x 1 ) has the properties of a pdf of one continuous type of random
variable. It is called theconditional pdfof the continuous type of random variable
X 2 , given that the continuous type of random variableX 1 has the valuex 1 .When
fX 2 (x 2 )>0, the conditional pdf of the continuous random variableX 1 ,giventhat
the continuous type of random variableX 2 has the valuex 2 , is defined by

fX 1 |X 2 (x 1 |x 2 )=

fX 1 ,X 2 (x 1 ,x 2 )

fX 2 (x 2 )

,fX 2 (x 2 )> 0.