= Solution
The <Bernoulli distribution> probability mass function is
$$
\Pr(Y_i=y_i)=p_i^{y_i}(1-p_i)^{1-y_i}=\exp\left\{y_i\log\frac{p_i}{1-p_i}+\log(1-p_i)\right\}.
$$
Writing $\theta_i=\log[p_i/(1-p_i)]$ gives $p_i=e^{\theta_i}/(1+e^{\theta_i})$ and $\log(1-p_i)=-\log(1+e^{\theta_i})$. Therefore the <exponential family> representation has
$$
\boxed{\theta_i=\operatorname{logit}(p_i)=\beta^Tx_i,\quad b(\theta)=\log(1+e^\theta),\quad\phi=1,\quad c(y,\phi)=0.}
$$
The <logit link> is the canonical link.
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