# Zero inflated Poisson Regression Model distribution function derivation

Is it possible to provide hints on how to get the probability distribution as shown below:

Additionally from a different online source, they mention the case about hte logistic submodel = the intercept of the logit model, how does that fact there help? (picture 2)

Consider two random variables Z and Y, such that $$Z \sim Bernoulli(1 - \pi)$$, $$Y | Z = 0$$ is $$0$$ with probability $$1$$, and $$Y | Z = 1 \sim Poisson(\lambda).$$ Then $$Y \sim ZIP(\pi, \lambda)$$.
Marginalize over $$Z$$ (it's an easy sum) to get the PMF.
• Var[Y] = Var[E[Y|Z]] + E[Var[Y|Z]], how do you evaluate the E[Y|Z] and Var[Y|Z] respectively? – cgeorge2000 Nov 28 '20 at 1:15