128 Multivariate DistributionsAlthough the converse of Theorem 2.5.2 is not true, the contrapositive is; i.e.,
ifρ =0thenX andY are dependent. For instance, in Example 2.5.1, since
ρ=0.816, we know that the random variablesX 1 andX 2 discussed in this example
are dependent. As discussed in Section 10.8, this contrapositive is often used in
Statistics.
Exercise 2.5.7 points out that in the proof of Theorem 2.5.1, the discriminant
of the polynomialh(v)is0ifandonlyifρ=±1. In that caseXandYare linear
functions of one another with probability one; although, as shown, the relationship is
degenerate. This suggests the following interesting question: Whenρdoes not have
one of its extreme values, is there a line in thexy-plane such that the probability
forX andY tends to be concentrated in a band about this line? Under certain
restrictive conditions this is, in fact, the case, and under those conditions we can
look uponρas a measure of the intensity of the concentration of the probability for
XandY about that line.
We summarize these thoughts in the next theorem. For notation, letf(x, y)
denote the joint pdf of two random variablesXandY and letf 1 (x)denotethe
marginal pdf ofX. Recall from Section 2.3 that the conditional pdf ofY,given
X=x,is
f 2 | 1 (y|x)=
f(x, y)
f 1 (x)at points wheref 1 (x)>0, and the conditional mean ofY,givenX=x,isgivenby
E(Y|x)=∫∞−∞yf 2 | 1 (y|x)dy=∫∞−∞yf(x, y)dyf 1 (x)
,when dealing with random variables of the continuous type. This conditional mean
ofY,givenX=x, is, of course, a function ofx,sayu(x). In a like vein, the
conditional mean ofX,givenY=y, is a function ofy,sayv(y).
In caseu(x) is a linear function ofx,sayu(x)=a+bx, we say the conditional
mean ofY is linear inx;orthatY has a linear conditional mean. Whenu(x)=
a+bx, the constantsaandbhave simple values which we show in the following
theorem.
Theorem 2.5.3. Suppose(X, Y)have a joint distribution with the variances ofX
andY finite and positive. Denote the means and variances ofXandY byμ 1 ,μ 2
andσ^21 ,σ^22 , respectively, and letρbe the correlation coefficient betweenXandY.If
E(Y|X)is linear inXthen
E(Y|X)=μ 2 +ρσ 2
σ 1(X−μ 1 ) (2.5.4)and
E(Var(Y|X)) =σ 22 (1−ρ^2 ). (2.5.5)Proof:The proof is given in the continuous case. The discrete case follows similarly