Paper 2, Section II, F

Probability | Part IA, 2013

Let XX be a random variable taking values in the non-negative integers, and let GG be the probability generating function of XX. Assuming GG is everywhere finite, show that

G(1)=μ and G(1)=σ2+μ2μG^{\prime}(1)=\mu \text { and } G^{\prime \prime}(1)=\sigma^{2}+\mu^{2}-\mu

where μ\mu is the mean of XX and σ2\sigma^{2} is its variance. [You may interchange differentiation and expectation without justification.]

Consider a branching process where individuals produce independent random numbers of offspring with the same distribution as XX. Let XnX_{n} be the number of individuals in the nn-th generation, and let GnG_{n} be the probability generating function of XnX_{n}. Explain carefully why


Assuming X0=1X_{0}=1, compute the mean of XnX_{n}. Show that

Var(Xn)=σ2μn1(μn1)μ1\operatorname{Var}\left(X_{n}\right)=\sigma^{2} \frac{\mu^{n-1}\left(\mu^{n}-1\right)}{\mu-1}

Suppose P(X=0)=3/7\mathbb{P}(X=0)=3 / 7 and P(X=3)=4/7\mathbb{P}(X=3)=4 / 7. Compute the probability that the population will eventually become extinct. You may use standard results on branching processes as long as they are clearly stated.

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