Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

2.6. Extension to Several Random Variables 137

pdf of the particular group ofkvariables, provided that the latter pdf is positive.
We remark that there are many other conditional probability density functions; for
instance, see Exercise 2.3.12.
Because a conditional pdf is the pdf of a certain number of random variables,
the expectation of a function of these random variables has been defined. To em-
phasize the fact that a conditional pdf is under consideration, such expectations
are called conditional expectations. For instance, the conditional expectation of
u(X 2 ,...,Xn), givenX 1 =x 1 , is, for random variables of the continuous type,
given by

E[u(X 2 ,...,Xn)|x 1 ]=

∫∞

−∞

···

∫∞

−∞

u(x 2 ,...,xn)f 2 ,...,n| 1 (x 2 ,...,xn|x 1 )dx 2 ···dxn

providedf 1 (x 1 )>0 and the integral converges (absolutely). A useful random
variable is given byh(X 1 )=E[u(X 2 ,...,Xn)|X 1 )].
The above discussion of marginal and conditional distributions generalizes to
random variables of the discrete type by using pmfs and summations instead of
integrals.
Let the random variablesX 1 ,X 2 ,...,Xnhave the joint pdff(x 1 ,x 2 ,...,xn)and
the marginal probability density functionsf 1 (x 1 ),f 2 (x 2 ),...,fn(xn), respectively.
The definition of the independence ofX 1 andX 2 is generalized to the mutual
independence ofX 1 ,X 2 ,...,Xnas follows: The random variablesX 1 ,X 2 ,...,Xn
are said to bemutually independentif and only if

f(x 1 ,x 2 ,...,xn)≡f 1 (x 1 )f 2 (x 2 )···fn(xn),

for the continuous case. In the discrete case,X 1 ,X 2 ,...,Xnare said to bemutually
independentif and only if

p(x 1 ,x 2 ,...,xn)≡p 1 (x 1 )p 2 (x 2 )···pn(xn).

SupposeX 1 ,X 2 ,...,Xnare mutually independent. Then

P(a 1 <X 1 <b 1 ,a 2 <X 2 <b 2 ,...,an<Xn<bn)

= P(a 1 <X 1 <b 1 )P(a 2 <X 2 <b 2 )···P(an<Xn<bn)

=

∏n

i=1

P(ai<Xi<bi),

where the symbol

∏n

i=1φ(i) is defined to be

∏n

i=1

φ(i)=φ(1)φ(2)···φ(n).

The theorem that

E[u(X 1 )v(X 2 )] =E[u(X 1 )]E[v(X 2 )]