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

probability distribution from wikipedia

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)

picture 2

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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.

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  • $\begingroup$ Just wondering how you go about showing the variance of Y: $\endgroup$ – cgeorge2000 Nov 28 '20 at 1:13
  • $\begingroup$ $$Var[Y] = Var[E[Y|Z]] + E[Var[Y|Z]], how do you evaluate the E[Y|Z] and Var[Y|Z] respectively? $\endgroup$ – cgeorge2000 Nov 28 '20 at 1:15
  • $\begingroup$ also is it possible to clarify Y|Z = 0 is 0 with probability 1?? @Jason $\endgroup$ – cgeorge2000 Nov 28 '20 at 3:02
  • $\begingroup$ Given that Z = 0, Y is 0 with probability 1, so the expected value is 0 and the variance is 0. Given that Z = 1, Y ~ Pois(lambda), so the expected value is lambda and the variance is lambda too. So now we can write E(Y | Z) = lambda * Z, and Var(Y | Z) as the same. Both are functions of Z. Now finding Var(Y) is a straightforward application of the formula you mentioned. $\endgroup$ – Jason Nov 30 '20 at 22:15
  • $\begingroup$ As for clarifying Y | Z = 0, I'm not sure what else there is to say. Given Z = 0, Y follows a point mass distribution located at 0. That is, for any omega in the sample space, Y(omega) = 0. $\endgroup$ – Jason Nov 30 '20 at 22:22

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