= Solution
For the <Poisson regression>, the <log-likelihood> is
$$
\ell(\beta)=\sum_{i=1}^n\left[Y_ix_i^T\beta-e^{x_i^T\beta}-\log(Y_i!)\right].
$$
Since $\mu_i=e^{x_i^T\beta}$, the <score function> and its derivative are
$$
U(\beta)=\sum_i x_i(Y_i-\mu_i)=X^T(Y-\mu),\qquad
\frac{\partial U}{\partial\beta^T}=-X^TWX,\quad W=\operatorname{diag}(\mu_i).
$$
Consequently a finite <maximum-likelihood estimator> satisfies
$$
\boxed{X^T(Y-\widehat\mu)=0,\qquad\widehat\mu_i=e^{x_i^T\widehat\beta}.}
$$
These nonlinear equations are normally solved by <iteratively reweighted least squares>. If $X$ has full column rank, the Hessian is negative definite and any finite solution is the unique maximizer. Full rank does not by itself rule out a boundary solution at infinite coefficients for every conceivable data set. The Poisson <generalized linear model> has dispersion one, so there is no separate scale parameter in this likelihood.
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