2.5. The Correlation Coefficient 133In each case compute the correlation coefficient ofXandY.2.5.2.LetXandYhave the joint pmf described as follows:(x, y) (1,1) (1,2) (1,3) (2,1) (2,2) (2,3)p(x, y) 152 154 153 151 151 154andp(x, y) is equal to zero elsewhere.(a)Find the meansμ 1 andμ 2 , the variancesσ^21 andσ 22 , and the correlation
coefficientρ.(b)ComputeE(Y|X=1),E(Y|X= 2), and the lineμ 2 +ρ(σ 2 /σ 1 )(x−μ 1 ). Do
the points [k, E(Y|X=k)],k=1,2, lie on this line?2.5.3.Letf(x, y)=2, 0 <x<y, 0 <y<1, zero elsewhere, be the joint pdf of
XandY. Show that the conditional means are, respectively, (1 +x)/ 2 , 0 <x<1,
andy/ 2 , 0 <y<1. Show that the correlation coefficient ofXandY isρ=^12.2.5.4.Show that the variance of the conditional distribution ofY,givenX=x,in
Exercise 2.5.3, is (1−x)^2 / 12 , 0 <x<1, and that the variance of the conditional
distribution ofX,givenY=y,isy^2 / 12 , 0 <y<1.
2.5.5.Verify the results of equations (2.5.11) of this section.2.5.6.LetXandY have the joint pdff(x, y)=1, −x<y<x, 0 <x<1,
zero elsewhere. Show that, on the set of positive probability density, the graph of
E(Y|x) is a straight line, whereas that ofE(X|y) is not a straight line.2.5.7.In the proof of Theorem 2.5.1, consider the case when the discriminant of
the polynomialh(v) is 0. Show that this is equivalent toρ=±1. Consider the case
whenρ= 1. Find the unique root ofh(v) and then use the fact thath(v)is0at
this root to show thatYis a linear function ofXwith probability 1.
2.5.8.Letψ(t 1 ,t 2 )=logM(t 1 ,t 2 ), whereM(t 1 ,t 2 )isthemgfofXandY. Show
that
∂ψ(0,0)
∂ti
,∂^2 ψ(0,0)
∂t^2 i,i=1, 2 ,and
∂^2 ψ(0,0)
∂t 1 ∂t 2
yield the means, the variances, and the covariance of the two random variables.
Use this result to find the means, the variances, and the covariance ofXandYof
Example 2.5.6.2.5.9.LetXandYhave the joint pmfp(x, y)=^17 ,(0,0),(1,0),(0,1),(1,1),(2,1),
(1,2),(2,2), zero elsewhere. Find the correlation coefficientρ.2.5.10.LetX 1 andX 2 have the joint pmf described by the following table: