### 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

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