Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

2.1. Distributions of Two Random Variables 99

2.1.8.LetXandYhave the pdff(x, y)=1, 0 <x< 1 , 0 <y<1, zero elsewhere.
Find the cdf and pdf of the productZ=XY.
2.1.9.Let 13 cards be taken, at random and without replacement, from an ordinary
deck of playing cards. IfXis the number of spades in these 13 cards, find the pmf of
X. If, in addition,Yis the number of hearts in these 13 cards, find the probability
P(X=2,Y= 5). What is the joint pmf ofXandY?

2.1.10.Let the random variablesX 1 andX 2 have the joint pmf described as follows:

(x 1 ,x 2 ) (0,0) (0,1) (0,2) (1,0) (1,1) (1,2)

p(x 1 ,x 2 ) 122 123 122 122 122 121

andp(x 1 ,x 2 ) is equal to zero elsewhere.

(a)Write these probabilities in a rectangular array as in Example 2.1.4, recording

each marginal pdf in the “margins.”

(b)What isP(X 1 +X 2 =1)?
2.1.11.LetX 1 andX 2 have the joint pdff(x 1 ,x 2 )=15x^21 x 2 , 0 <x 1 <x 2 <1,
zero elsewhere. Find the marginal pdfs and computeP(X 1 +X 2 ≤1).
Hint: Graph the spaceX 1 andX 2 and carefully choose the limits of integration
in determining each marginal pdf.
2.1.12.LetX 1 ,X 2 be two random variables with the joint pmfp(x 1 ,x 2 ), (x 1 ,x 2 )∈
S,whereSis the support ofX 1 ,X 2 .LetY=g(X 1 ,X 2 ) be a function such that
∑∑

(x 1 ,x 2 )∈S

|g(x 1 ,x 2 )|p(x 1 ,x 2 )<∞.

By following the proof of Theorem 1.8.1, show that

E(Y)=

∑∑

(x 1 ,x 2 )∈S

g(x 1 ,x 2 )p(x 1 ,x 2 ).

2.1.13.LetX 1 ,X 2 be two random variables with the joint pmfp(x 1 ,x 2 )=(x 1 +
x 2 )/ 12 ,forx 1 =1, 2 ,x 2 =1,2, zero elsewhere. ComputeE(X 1 ),E(X 12 ),E(X 2 ),
E(X^22 ), andE(X 1 X 2 ). IsE(X 1 X 2 )=E(X 1 )E(X 2 )? FindE(2X 1 − 6 X 22 +7X 1 X 2 ).
2.1.14.LetX 1 ,X 2 be two random variables with joint pdff(x 1 ,x 2 )=4x 1 x 2 ,
0 <x 1 <1, 0<x 2 < 1 ,zero elsewhere. ComputeE(X 1 ),E(X 12 ),E(X 2 ),E(X^22 ),
andE(X 1 X 2 ). IsE(X 1 X 2 )=E(X 1 )E(X 2 )? FindE(3X 2 − 2 X 12 +6X 1 X 2 ).

2.1.15.LetX 1 ,X 2 be two random variables with joint pmfp(x 1 ,x 2 )=(1/2)x^1 +x^2 ,
for 1≤xi<∞,i=1, 2 ,wherex 1 andx 2 are integers, zero elsewhere. Determine
the joint mgf ofX 1 ,X 2. Show thatM(t 1 ,t 2 )=M(t 1 ,0)M(0,t 2 ).

2.1.16.LetX 1 ,X 2 be two random variables with joint pdff(x 1 ,x 2 )=x 1 exp{−x 2 },
for 0<x 1 <x 2 <∞, zero elsewhere. Determine the joint mgf ofX 1 ,X 2 .Does
M(t 1 ,t 2 )=M(t 1 ,0)M(0,t 2 )?

2.1.17.LetXandYhave the joint pdff(x, y)=6(1−x−y),x+y< 1 , 0 <x,

0 <y, zero elsewhere. ComputeP(2X+3Y<1) andE(XY+2X^2 ).