= Solution
A <generalized linear model> has independent responses whose distributions belong to an <exponential family>:
$$
f_i(y_i;\theta_i,\phi)=\exp\left\{\frac{y_i\theta_i-b(\theta_i)}{a_i(\phi)}+c_i(y_i,\phi)\right\}.
$$
Here the <expected value> and <variance> are $\mu_i=b'(\theta_i)$ and $\operatorname{Var}(Y_i)=a_i(\phi)b''(\theta_i)$. A specified one-to-one <link function> $g$ connects the mean to the <linear predictor>: $g(\mu_i)=x_i^T\beta$. The <canonical link function> is $g(\mu_i)=\theta_i$.
In <binomial regression>, $Y_i\sim\operatorname{Bin}(m_i,p_i)$ independently, with known trial counts $m_i$ and probabilities related to the predictor. The <logistic regression> and <probit regression> specifications are respectively
$$
\boxed{\log\frac{p_i}{1-p_i}=x_i^T\beta,\quad p_i=\frac{e^{x_i^T\beta}}{1+e^{x_i^T\beta}};\qquad \Phi^{-1}(p_i)=x_i^T\beta,\quad p_i=\Phi(x_i^T\beta).}
$$
Here $\Phi$ is the standard <normal distribution> function. The binomial log-density contains $y_i\log(p_i/(1-p_i))+m_i\log(1-p_i)$, so its natural parameter is the log-odds: the logistic link is canonical. If the response mean is written as $\mu_i=m_ip_i$, the canonical link is $g_i(\mu_i)=\log(\mu_i/(m_i-\mu_i))$.
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