4c: We here assume that I and X follow the same distributions as in 4b.
We consider the process Z n , where Z 0 = 1 , and at each generation n 1 , Z n + 1 = I n + i = 1 Z n X n , i , where the X n , i 's are i.i.d. with common distribution X , and the I n 's are i.i.d and independent from the X n , i 's, with common distribution I (this can be seen as a branching process where an immigrant population arrives at each generation and independently reproduces at the next one). a. Find the generating function of Z n , as a function of the generating functions of I , X , and n . b. Let X Binomial ( 1 , p ) and I Poisson ( ) , where 0 < p < 1 and > 0 . Find the generating function of Z n as a function of , p and n . c (bonus). Find the expectation and variance of Z n . Remark: Using the so-called "continuity theorem" that states that convergence for the generating function implies convergence in distribution, the result from question b allows to show that the limiting distribution of the process is Poisson, which you can also verify by running this week's notebook (link).
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4c- We here assume that I and X follow the same distributions as in 4b.docx
1. 4c: We here assume that I and X follow the same distributions as in 4b.
We consider the process Z n , where Z 0 = 1 , and at each generation n 1 , Z n + 1 = I n + i = 1 Z
n X n , i , where the X n , i 's are i.i.d. with common distribution X , and the I n 's are i.i.d and
independent from the X n , i 's, with common distribution I (this can be seen as a branching
process where an immigrant population arrives at each generation and independently reproduces
at the next one). a. Find the generating function of Z n , as a function of the generating functions
of I , X , and n . b. Let X Binomial ( 1 , p ) and I Poisson ( ) , where 0 < p < 1 and > 0 . Find the
generating function of Z n as a function of , p and n . c (bonus). Find the expectation and
variance of Z n . Remark: Using the so-called "continuity theorem" that states that convergence
for the generating function implies convergence in distribution, the result from question b allows
to show that the limiting distribution of the process is Poisson, which you can also verify by
running this week's notebook (link).
