For the exponential family of distributions defined in (4.41), show

(a) \(E(y)=\frac{d}{d \theta} b(\theta)\).

(b) \(\operatorname{Var}(y)=a(\phi) \frac{d^{2}}{d \theta^{2}} b(\theta)\).

(c) Assume that \(y \sim N\left(\mu, \sigma^{2}\right)\) and \(\sigma^{2}\) is known. Show that the canonical link for the mean \(\mu\) is the identity function \(g(\theta)=\theta\).

(d) Show that the canonical link for Bernoulli distribution is the logistic function.

Transcribed Image Text:

f (y | 0) = exp [(y-b (0))/a (o)+c (y, 0)], (4.41)