(i) Suppose that the random variable $Y$ has density function of the form

$$f(y \mid \theta, \phi)=\exp \left[\frac{y \theta-b(\theta)}{\phi}+c(y, \phi)\right]$$

where $\phi>0$. Show that $Y$ has expectation $b^{\prime}(\theta)$ and variance $\phi b^{\prime \prime}(\theta)$.

(ii) Suppose now that $Y_{1}, \ldots, Y_{n}$ are independent negative exponential variables, with $Y_{i}$ having density function $f\left(y_{i} \mid \mu_{i}\right)=\frac{1}{\mu_{i}} e^{-y_{i} / \mu_{i}}$ for $y_{i}>0$. Suppose further that $g\left(\mu_{i}\right)=\beta^{T} x_{i}$ for $1 \leqslant i \leqslant n$, where $g(\cdot)$ is a known 'link' function, and $x_{1}, \ldots, x_{n}$ are given covariate vectors, each of dimension $p$. Discuss carefully the problem of finding $\hat{\beta}$, the maximum-likelihood estimator of $\beta$, firstly for the case $g\left(\mu_{i}\right)=1 / \mu_{i}$, and secondly for the case $g(\mu)=\log \mu_{i}$; in both cases you should state the large-sample distribution of $\hat{\beta}$.

[Any standard theorems used need not be proved.]