Given
\[(Y_1, X_1), (Y_2, X_2),\dots,(Y_n, X_n) \sim F_{Y, X}\]
It is assumed that the regression function has this form
\[r(x) = E[Y|X=x] = g^{-1}(\beta_0 + \beta_1x) = g^{-1}(\eta)\]
\[g(E[Y|X=x]) = \beta_0 + \beta_1x\]
where \(Y\) is a random variable from a particular distribution of the exponential family, \(g\) is called the link function and \(\eta\) the linear predictor.
So, the model is completely specified given:
The probability distribution of \(Y\).
The relation between the expected values of \(Y\) and a linear combination of the explanatory variables.
\[r(\textbf{x}) = E[Y|\textbf{X}=\textbf{x}] = g^{-1}(\beta_1 + \beta_2x_2 + \dotsc + \beta_k x_k ) = g^{-1}(\eta)\]