4.12 Poisson Regression Model mean (lambda)

“[R]ather than modeling [a count response variable], \(Y\), as a Poisson distribution with a fixed mean value like \(\lambda\) = 5, we would like to allow the mean to vary as a function of the covariates.”

The mean \(\lambda\) can be modeled as a function of the predictor variables as follows:

\(log(\lambda(X_1, ..., X_p) = \beta_0 + \beta_1X_1 + ... + \beta_pX_p\)

NB: taking the log ensures that \(\lambda\) can only be non-negative.

This is equivalent to representing the mean \(\lambda\) as follows:

\(\lambda = \text{E}(Y) = \lambda(X_1, ..., X_p) = e^{\beta_0 + \beta_1X_1 + ... + \beta_pX_p}\)