I have had the question below on my term test a few weeks ago; however, I do not know, and I cannot find what textbook this has come from. If could you let me know where that came from, it would be VERY highly appreciated.
The question is as follows:
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Assume that the random variable U admits the following compound Poisson representation:
$$U= \sum_{i=1}^{N} B_i$$
Here, the random variable N is Poisson-Distributed with mean $\eta$, and {$B_n, n \geq1$} are independent and identically distributed random variables with common Bernoulli 0/1 distribution with the probability of success $p$, such that $P\{B_i=1\}=p$. Suppose that the random variables {$B_n, n \geq1$} do not depend on the Poisson counting random variable $N$
(i) Find the expectation of the variable $U$
(ii) Find the variance of the random variable $U$
(iii) Compare your results from (i) and (ii). Does it give you a hint on what the distribution of the random variable $U$ can be?