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

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