2.II.10F

There is a random number $N$ of foreign objects in my soup, with mean $\mu$ and finite variance. Each object is a fly with probability $p$, and otherwise is a spider; different objects have independent types. Let $F$ be the number of flies and $S$ the number of spiders.

(a) Show that $G_{F}(s)=G_{N}(p s+1-p) .\left[G_{X}\right.$ denotes the probability generating function of a random variable $X$. You should present a clear statement of any general result used.]

(b) Suppose $N$ has the Poisson distribution with parameter $\mu$. Show that $F$ has the Poisson distribution with parameter $\mu p$, and that $F$ and $S$ are independent.

(c) Let $p=\frac{1}{2}$ and suppose that $F$ and $S$ are independent. [You are given nothing about the distribution of $N$.] Show that $G_{N}(s)=G_{N}\left(\frac{1}{2}(1+s)\right)^{2}$. By working with the function $H(s)=G_{N}(1-s)$ or otherwise, deduce that $N$ has the Poisson distribution. [You may assume that $\left(1+\frac{x}{n}+\mathrm{o}\left(n^{-1}\right)\right)^{n} \rightarrow e^{x}$ as $n \rightarrow \infty$.]

*Typos? Please submit corrections to this page on GitHub.*