Given that Y follows a distribution in the exponential family of form in (8.2), show that E(Y ) = b 0 (θ) and Var(Y ) = b 00(θ)a(φ) by using the following two general likelihood results E µ ∂L ∂θ ¶ = 0 and − E µ ∂ 2L ∂θ2 ¶ = E µ ∂L ∂θ ¶2 ,

(a) Class Level Information Class Value Design Variables Eth N 1 A 0 Sex M 1 F 0 Age F3 1 0 0 F2 0 1 0 F1 0 0 1 F0 0 0 0 Lrn SL 1 AL 0

(b) Analysis of Parameter Estimates Standard Wald 95% ChiParameter DF Estimate Error Confidence Limits Square Intercept 1 2.7154 0.0647 2.5886 2.8422 1762.30 Eth N 1 −0.5336 0.0419 −0.6157 −0.4515 162.32 Sex M 1 0.1616 0.0425 0.0782 0.2450 14.43 Age F3 1 0.4277 0.0677 0.2950 0.5604 39.93 F2 1 0.2578 0.0624 0.1355 0.3802 17.06 F1 1 −0.3339 0.0701 −0.4713 −0.1965 22.69 Lrn SL 1 0.3489 0.0520 0.2469 0.4509 44.96 where L(θ, φ) = log fY (y) = {yθ − b(θ)}/a(φ) + c(y, φ) is the log likelihood. The two equations in (8.32) hold under some regularity conditions given in Sec. 4.8 of Cox and Hinkley (1974).