What is the distribution of the error term in the Poisson Regression model? Given a Poisson regression model as $y = E(y\mid x) + ε$ where $λ = E(y\mid x) = \exp(x'β)$ with $y$ from the Poisson distribution ($\operatorname{Poisson}(λ)$) I am trying to understand the distribution of $ε$. I tried writing the error term as $ε = y - E(y\mid x) = y - λ$ and obtained the PMF of $ε$ as $P(ε = k) = P(y = k+λ) = {λ^{k+λ}e^{-λ}}/{(k+λ)!}$. So can it be concluded that the error term of the Poisson model follow a distribution with the obtained PMF or there is a known distribution that the error term follows?

@whuber's comment is the answer.  Typically we don't assume an independent error term with non-normal generalized linear models.  The response distribution already contains the theorized stochastic elements of the situation.  In the context of a Poisson GLM, we assume that the expected value of the inverse transformed linear predictor (the most common link would be the log) is equal to the parameter ($\lambda$) that controls the behavior of the conditional response distribution.  The observed datum is seen as a realized value drawn from that distribution.
