Inbivariatemodeling,supposewetreat X also asarandomvariable,with (X,Y ) havinga bivariatenormaldistributionwithcorrelation . Regressionanalysisestimatestheequation E(Y S X = x) = Y

Inbivariatemodeling,supposewetreat X also asarandomvariable,with (X,Y ) havinga bivariatenormaldistributionwithcorrelation ρ. Regressionanalysisestimatestheequation E(Y S X = x) = μY + ρ(σY ~σX)‰x − μX).

(a) Statetheequationestimatedifweinsteaduse Y to predict X.

(b) If E(Y ) = β0 + β1x and E(X) = β∗

0 + β∗

1 y, showthatthetwoseparateregressionssatisfy

β1~(1~β∗

1 ) = ρ2, so β1 = 1~β∗

1 only when SρS = 1.