= Solution
The <Poisson distribution> mass function is
$$
P(Y=y)=\frac{e^{-\mu}\mu^y}{y!}
=\exp\{y\log\mu-\mu-\log(y!)\},\qquad y=0,1,\ldots.
$$
It is an <exponential dispersion family of order one> with
$$
\boxed{\theta=\log\mu,\qquad b(\theta)=e^\theta,\qquad\phi=1,\qquad c(y,\phi)=-\log(y!).}
$$
Indeed, $b'(\theta)=e^\theta=E(Y)$ and $\phi b''(\theta)=e^\theta=\operatorname{Var}(Y)$. A <generalized linear model> specifies <independent random variables> from a response <exponential family>, together with a <linear predictor> and a <link function> relating it to the response mean. The <Poisson canonical link> is $g(\mu)=\log\mu$.
For the insurance counts, let class $1$ and merit $0$ be the <reference levels in a regression factor>. The fitted <Poisson regression> assumes
$$
Y_{ij}\overset{\mathrm{ind}}\sim\operatorname{Pois}(\mu_{ij}),\qquad
\mu_{ij}=n_{ij}\lambda_{ij},\qquad
\log\lambda_{ij}=\alpha+c_i+m_j,\qquad c_1=m_0=0.
$$
The insured exposure $n_{ij}$ is known; its <logarithm> is an <offset>, not a coefficient to estimate. The model has eight free mean parameters and fixes the <dispersion parameter> at one. It assumes additive class and merit effects on the log-rate scale, hence multiplicative effects on the rate scale without a class-by-merit <interaction>.
Apart from constants independent of the parameters, the <log-likelihood> is
$$
\ell=\sum_{i,j}\left[Y_{ij}(\alpha+c_i+m_j)-n_{ij}e^{\alpha+c_i+m_j}\right].
$$
Differentiating gives the <Poisson regression margin-matching score equations>
$$
\sum_{i,j}(Y_{ij}-\widehat\mu_{ij})=0,\qquad
\sum_i(Y_{ij}-\widehat\mu_{ij})=0\quad(j=1,2,3),\qquad
\sum_j(Y_{ij}-\widehat\mu_{ij})=0\quad(i=2,3,4,5).
$$
Thus the fitted totals equal the observed totals in each class and merit category, including the reference categories by subtraction. In <design matrix> notation these equations are $Z^T(Y-\widehat\mu)=0$. <Fisher scoring> solves them iteratively; the reported three iterations describe the numerical fit, not an additional statistical result.
The <regression intercept> gives the baseline fitted claim rate $e^{-2.0357359}=0.130584$ per insured car year. The merit <rate ratios>, relative to merit $0$ at the same class, are
$$
\boxed{e^{\widehat m_1}=0.87131,\qquad e^{\widehat m_2}=0.80197,\qquad e^{\widehat m_3}=0.61082.}
$$
The class <rate ratios>, relative to class $1$ at the same merit, are
$$
\boxed{e^{\widehat c_2}=1.34963,\quad e^{\widehat c_3}=1.59848,\quad
e^{\widehat c_4}=1.69190,\quad e^{\widehat c_5}=1.24054.}
$$
Each predicted cell rate is the baseline rate times its class and merit multipliers. The output's coefficient-to-<standard error> ratios are asymptotic normal <Wald test> statistics under the fitted <Poisson regression>, despite the software label “t value”; there is no estimated Gaussian residual scale in this model. All nonreference effects have very large absolute ratios.
The null <Poisson deviance> is $33854.16$ on nineteen <statistical degrees of freedom>; adding the seven factor coefficients reduces it by $33274.6437$. But the final <Poisson deviance> is still $579.5163$ on twelve <statistical degrees of freedom>, and the <deviance residuals> include values near $-10.79$ and $11.63$. These are far beyond what a well-fitting unit-dispersion model would normally produce. \b[The additive Poisson model fits substantially better than a common rate, but remains grossly inadequate.] A class-by-merit <interaction>, <overdispersion>, or <statistical dependence> between claims could contribute; the summary alone does not distinguish these explanations. Its very small nominal <standard errors> and formal <Wald tests> rely on the rejected model, so improved mean structure and error assumptions are needed before treating them as reliable uncertainty assessments.
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