Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

2.5. The Correlation Coefficient 129

by changing integrals to sums. LetE(Y|x)=a+bx.From

E(Y|x)=

∫∞

−∞

yf(x, y)dy

f 1 (x)

=a+bx,

we have ∫∞

−∞

yf(x, y)dy=(a+bx)f 1 (x). (2.5.6)

If both members of Equation (2.5.6) are integrated onx, it is seen that

E(Y)=a+bE(X)

or
μ 2 =a+bμ 1 , (2.5.7)
whereμ 1 =E(X)andμ 2 =E(Y). If both members of Equation (2.5.6) are first
multiplied byxandthenintegratedonx,wehave

E(XY)=aE(X)+bE(X^2 ),

or
ρσ 1 σ 2 +μ 1 μ 2 =aμ 1 +b(σ 12 +μ^21 ), (2.5.8)

whereρσ 1 σ 2 is the covariance ofXandY. The simultaneous solution of equations
(2.5.7) and (2.5.8) yields

a=μ 2 −ρ

σ 2

σ 1

μ 1 and b=ρ

σ 2

σ 1

.

These values give the first result (2.5.4).
Next, the conditional variance ofYis given by

Var(Y|x)=

∫∞

−∞

[

y−μ 2 −ρ

σ 2

σ 1

(x−μ 1 )

] 2

f 2 | 1 (y|x)dy

=

∫∞

−∞

[

(y−μ 2 )−ρ

σ 2

σ 1

(x−μ 1 )

] 2

f(x, y)dy

f 1 (x)

. (2.5.9)

This variance is nonnegative and is at most a function ofxalone. If it is multiplied
byf 1 (x) and integrated onx, the result obtained is nonnegative. This result is
∫∞

−∞

∫∞

−∞

[

(y−μ 2 )−ρ

σ 2

σ 1

(x−μ 1 )

] 2

f(x, y)dydx

=

∫∞

−∞

∫∞

−∞

[

(y−μ 2 )^2 − 2 ρ

σ 2

σ 1

(y−μ 2 )(x−μ 1 )+ρ^2

σ^22

σ^21

(x−μ 1 )^2

]

f(x, y)dydx

= E[(Y−μ 2 )^2 ]− 2 ρ

σ 2

σ 1

E[(X−μ 1 )(Y−μ 2 )] +ρ^2

σ 22

σ 12

E[(X−μ 1 )^2 ]

= σ^22 − 2 ρ

σ 2

σ 1

ρσ 1 σ 2 +ρ^2

σ^22

σ^21

σ^21

= σ^22 − 2 ρ^2 σ 22 +ρ^2 σ^22 =σ 22 (1−ρ^2 ),