(i) Suppose that $Y_{1}, \ldots, Y_{n}$ are independent random variables, and that $Y_{i}$ has probability density function

$$f\left(y_{i} \mid \theta_{i}, \phi\right)=\exp \left[\left(y_{i} \theta_{i}-b\left(\theta_{i}\right)\right) / \phi+c\left(y_{i}, \phi\right)\right]$$

Assume that $E\left(Y_{i}\right)=\mu_{i}$, and that $g\left(\mu_{i}\right)=\beta^{T} x_{i}$, where $g(\cdot)$ is a known 'link' function, $x_{1}, \ldots, x_{n}$ are known covariates, and $\beta$ is an unknown vector. Show that

$$\mathbb{E}\left(Y_{i}\right)=b^{\prime}\left(\theta_{i}\right), \operatorname{var}\left(Y_{i}\right)=\phi b^{\prime \prime}\left(\theta_{i}\right)=V_{i} \text {, say, }$$

and hence

$$\frac{\partial l}{\partial \beta}=\sum_{i=1}^{n} \frac{\left(y_{i}-\mu_{i}\right) x_{i}}{g^{\prime}\left(\mu_{i}\right) V_{i}}, \text { where } l=l(\beta, \phi) \text { is the log-likelihood. }$$

(ii) The table below shows the number of train miles (in millions) and the number of collisions involving British Rail passenger trains between 1970 and 1984 . Give a detailed interpretation of the $R$ output that is shown under this table:

$\begin{array}{llll} & \text { year } & \text { collisions } & \text { miles } \\ 1 & 1970 & 3 & 281 \\ 2 & 1971 & 6 & 276 \\ 3 & 1972 & 4 & 268 \\ 4 & 1973 & 7 & 269 \\ 5 & 1974 & 6 & 281 \\ 6 & 1975 & 2 & 271 \\ 7 & 1976 & 2 & 265 \\ 8 & 1977 & 4 & 264 \\ 9 & 1978 & 1 & 267 \\ 10 & 1979 & 7 & 265 \\ 11 & 1980 & 3 & 267 \\ 12 & 1981 & 5 & 260 \\ 13 & 1982 & 6 & 231 \\ 14 & 1983 & 1 & 249\end{array}$

Call:

glm(formula $=$ collisions $\sim$ year $+\log ($ miles $)$, family $=$ poisson)

Coefficients:

$\begin{array}{lrrrr} & \text { Estimate } & \text { Std. Error } & \text { z value } & \operatorname{Pr}(>|z|) \\ \text { (Intercept) } & 127.14453 & 121.37796 & 1.048 & 0.295 \\ \text { year } & -0.05398 & 0.05175 & -1.043 & 0.297 \\ \log \text { (miles) } & -3.41654 & 4.18616 & -0.816 & 0.414\end{array}$

(Dispersion parameter for poisson family taken to be 1)

Null deviance: $15.937$ on 13 degrees of freedom

Residual deviance: $14.843$ on 11 degrees of freedom

Number of Fisher Scoring iterations: 4

Part II