Q1. Moments of exponential family distributions

Show that the exponential family distributions have moments \[\begin{eqnarray*} \mathbb{E}Y &=& \mu = b'(\theta) \\ \operatorname{Var}Y &=& \sigma^2 = b''(\theta) a(\phi). \end{eqnarray*}\]

Q2. Score and information matrix of GLM

Derive the gradient (score), negative Hessian, and Fisher information matrix (expected negative Hessian) of GLM.

Q3. ELMR Exercise 8.1 (p171)

Q4. ELMR Exercise 8.4 (p172)

Q5. ELMR Exercise 8.5 (p172)