Suppose that $Y_{1}, \ldots, Y_{n}$ are independent observations, with $Y_{i}$ having probability density function of the following form

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

where $\mathbb{E}\left(Y_{i}\right)=\mu_{i}$ and $g\left(\mu_{i}\right)=\beta^{T} x_{i}$. You should assume that $g()$ is a known function, and $\beta, \phi$ are unknown parameters, with $\phi>0$, and also $x_{1}, \ldots, x_{n}$ are given linearly independent covariate vectors. Show that

$$\frac{\partial \ell}{\partial \beta}=\sum \frac{\left(y_{i}-\beta_{i}\right)}{g^{\prime}\left(\mu_{i}\right) V_{i}} x_{i}$$

where $\ell$ is the log-likelihood and $V_{i}=\operatorname{var}\left(Y_{i}\right)=\phi b^{\prime \prime}\left(\theta_{i}\right)$.

Discuss carefully the (slightly edited) $\mathrm{R}$ output given below, and briefly suggest another possible method of analysis using the function $\mathrm{glm}$ ( ).

$>s<-\operatorname{scan}()$

1: $\begin{array}{llllll}33 & 63 & 157 & 38 & 108 & 159\end{array}$

7:

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$>r<-\operatorname{scan}()$

1: 327172565065248688773520

$7:$

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$>$ gender <- $\operatorname{scan}(, " \|)$

1: b b b g g g

$7:$

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$>$ age <- $\operatorname{scan}(, " \prime)$

1: 13&under 14-18 19&over

4: 13&under 14-18 19&over

7 :

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$>$ gender <- factor (gender) ; age <- factor (age)

$>\operatorname{summary}(\mathrm{glm}(\mathrm{s} / \mathrm{r} \sim$ gender $+$ age, binomial, weights $=\mathrm{r}))$

Coefficients:

Null deviance: $221.797542$ on 5 degrees of freedom

Residual deviance: $0.098749$ on 2 degrees of freedom

Number of Fisher Scoring iterations: 3