Definition for a single predictor

\[r(x) = E[Y|X=x] = g^{-1}(\beta_0 + \beta_1x) = g^{-1}(\eta)\] that can be also written as \[g(E[Y|X=x]) = \beta_0 + \beta_1x\]

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.

Definition for multiple predictors

\[r(\textbf{x}) = E[Y|\textbf{X}=\textbf{x}] = g^{-1}(\beta_1 + \beta_2x_2 + \dotsc + \beta_k x_k ) = g^{-1}(\eta)\]