Show that the exponential family distributions have moments \[\begin{eqnarray*} \mathbb{E}Y &=& \mu = b'(\theta) \\ \operatorname{Var}Y &=& \sigma^2 = b''(\theta) a(\phi). \end{eqnarray*}\]
Derive the gradient (score), negative Hessian, and Fisher information matrix (expected negative Hessian) of GLM.