Suppose that \(Y\) is Poisson distributed with mean \(\mu\).

(a) Verify that the first four moments are, respectively, \(\mu, \mu, \mu\), and \(3 \mu^{2}+\mu\).

(b) Show that there is a linear relationship between \(\operatorname{Pr}[Y=j]\) and \(\operatorname{Pr}[Y=j-\) 1], \(j=1,2, \ldots\).

(c) Consider the Poisson MLE in the regression case with \(\mu_{i}=\exp \left(\mathbf{x}_{i}^{\prime} \beta\right)\). Possible estimates of the variance of the Poisson MLE include \(\widehat{V}[\beta]=\) \(\left[\sum_{i} \widehat{\mu}_{i} \mathbf{x}_{i} \mathbf{x}_{i}^{\prime}\right]^{-1}\) and \(\widetilde{V}[\widehat{\beta}]=\left[\sum_{i}\left(y_{i}-\widehat{\mu}_{i}\right)^{2} \mathbf{x}_{i} \mathbf{x}_{i}^{\prime}\right]^{-1}\). Show that they are asymptotically equivalent (upon scaling by \(N\) ) if the data density is correctly specified.