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:

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