### Definition

$$Y$$ is distributed $$Bernoulli(p)$$ and

$g^{-1}(\eta) = \frac{e^\eta}{1 + e^\eta}$ That is,

$r(x)=E[Y|X=x]=p(x)=\frac{e^{\beta_0 + \beta_1x}}{1 + e^{\beta_0 + \beta_1x}}$ given that the $E[Y] = p$

for $$Bernoulli(p)$$.

So, we can also write

$g(E[Y|X=x]) = log \frac{p}{1-p}= \beta_0 + \beta_1x$