The question:

Suppose $Y$ is discrete and only takes on non-negative integers and that the conditional distribution of $Y$ given $X=x$ is Poisson, that is, $$P(Y=y|X=x) = \frac{\exp(-x'\beta) (x'\beta)^y}{y!}$$ where $y = 0, 1, 2, \cdots$. First compute $E(Y|X=x)$ and $Var(Y|X=x)$, does this justify a linear regression model of the form $y = x'\beta + e$?

My attempt:

I have calculated $E(Y|X=x) = Var(Y|X=x) = x'\beta$ by the properties of a Poisson distribution. I am unsure how to answer the last part of the question related to the linear regression model. Any help would be appreciated.