116 Multivariate Distributions2.3.2.Letf 1 | 2 (x 1 |x 2 )=c 1 x 1 /x^22 , 0 <x 1 <x 2 , 0 <x 2 <1, zero elsewhere, and
f 2 (x 2 )=c 2 x^42 , 0 <x 2 <1, zero elsewhere, denote, respectively, the conditional pdf
ofX 1 ,givenX 2 =x 2 , and the marginal pdf ofX 2. Determine:
(a)The constantsc 1 andc 2.(b)The joint pdf ofX 1 andX 2.(c)P(^14 <X 1 <^12 |X 2 =^58 ).(d)P(^14 <X 1 <^12 ).2.3.3.Letf(x 1 ,x 2 )=21x^21 x^32 , 0 <x 1 <x 2 <1, zero elsewhere, be the joint pdf
ofX 1 andX 2.
(a)Find the conditional mean and variance ofX 1 ,givenX 2 =x 2 , 0 <x 2 <1.(b)Find the distribution ofY=E(X 1 |X 2 ).(c)DetermineE(Y)andVar(Y) and compare these toE(X 1 )andVar(X 1 ), re-
spectively.2.3.4.SupposeX 1 andX 2 are random variables of the discrete type that have
the joint pmfp(x 1 ,x 2 )=(x 1 +2x 2 )/ 18 ,(x 1 ,x 2 )=(1,1),(1,2),(2,1),(2,2), zero
elsewhere. Determine the conditional mean and variance ofX 2 ,givenX 1 =x 1 ,for
x 1 = 1 or 2. Also, computeE(3X 1 − 2 X 2 ).
2.3.5.LetX 1 andX 2 be two random variables such that the conditional distribu-
tions and means exist. Show that:
(a)E(X 1 +X 2 |X 2 )=E(X 1 |X 2 )+X 2 ,(b)E(u(X 2 )|X 2 )=u(X 2 ).2.3.6.LetthejointpdfofXandYbe given by
f(x, y)={ 2
(1+x+y)^30 <x<∞,^0 <y<∞
0elsewhere.(a)Compute the marginal pdf ofXand the conditional pdf ofY,givenX=x.(b)For a fixedX=x, computeE(1 +x+Y|x) and use the result to compute
E(Y|x).2.3.7.SupposeX 1 andX 2 are discrete random variables which have the joint pmf
p(x 1 ,x 2 )=(3x 1 +x 2 )/ 24 ,(x 1 ,x 2 )=(1,1),(1,2),(2,1),(2,2), zero elsewhere. Find
the conditional meanE(X 2 |x 1 ), whenx 1 =1.
2.3.8.LetXandYhave the joint pdff(x, y)=2exp{−(x+y)}, 0 <x<y<∞,
zero elsewhere. Find the conditional meanE(Y|x)ofY,givenX=x.