Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

2.7. Transformations for Several Random Variables 143

2.6.4.A fair die is cast at random three independent times. Let the random variable

Xibe equal to the number of spots that appear on theith trial,i=1, 2 ,3. Let the

random variableY be equal to max(Xi). Find the cdf and the pmf ofY.

Hint:P(Y≤y)=P(Xi≤y, i=1, 2 ,3).

2.6.5.LetM(t 1 ,t 2 ,t 3 ) be the mgf of the random variablesX 1 ,X 2 ,andX 3 of
Bernstein’s example, described in the remark following Example 2.6.2. Show that

M(t 1 ,t 2 ,0) =M(t 1 , 0 ,0)M(0,t 2 ,0),M(t 1 , 0 ,t 3 )=M(t 1 , 0 ,0)M(0, 0 ,t 3 ),

and
M(0,t 2 ,t 3 )=M(0,t 2 ,0)M(0, 0 ,t 3 )

are true, but that

M(t 1 ,t 2 ,t 3 ) =M(t 1 , 0 ,0)M(0,t 2 ,0)M(0, 0 ,t 3 ).

ThusX 1 ,X 2 ,X 3 are pairwise independent but not mutually independent.

2.6.6.LetX 1 ,X 2 ,andX 3 be three random variables with means, variances, and
correlation coefficients, denoted byμ 1 ,μ 2 ,μ 3 ;σ 12 ,σ^22 ,σ^23 ;andρ 12 ,ρ 13 ,ρ 23 , respec-
tively. For constantsb 2 andb 3 , supposeE(X 1 −μ 1 |x 2 ,x 3 )=b 2 (x 2 −μ 2 )+b 3 (x 3 −μ 3 ).
Determineb 2 andb 3 in terms of the variances and the correlation coefficients.

2.6.7.Prove Corollary 2.6.1.

2.6.8.LetX=(X 1 ,...,Xn)′be ann-dimensional random vector, with the variance-
covariance matrix given in display (2.6.13). Show that theith diagonal entry of
Cov(X)isσ^2 i=Var(Xi) and that the (i, j)th off diagonal entry is Cov(Xi,Xj).

2.6.9.LetX 1 ,X 2 ,X 3 be iid with common pdff(x)=exp(−x), 0 <x<∞, zero
elsewhere. Evaluate:

(a)P(X 1 <X 2 |X 1 < 2 X 2 ).

(b)P(X 1 <X 2 <X 3 |X 3 <1).

2.7 Transformations for Several Random Variables

In Section 2.2 it was seen that the determination of the joint pdf of two functions of
two random variables of the continuous type was essentially a corollary to a theorem
in analysis having to do with the change of variables in a twofold integral. This
theorem has a natural extension ton-fold integrals. This extension is as follows.
Consider an integral of the form
∫
···

∫

A

f(x 1 ,x 2 ,...,xn)dx 1 dx 2 ···dxn

taken over a subsetAof ann-dimensional spaceS.Let

y 1 =u 1 (x 1 ,x 2 ,...,xn),y 2 =u 2 (x 1 ,x 2 ,...,xn),...,yn=un(x 1 ,x 2 ,...,xn),