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Calculus and Beyond Homework Help
Branching process, inductive proof
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[QUOTE="spitz, post: 3869998, member: 368691"] [h2]Homework Statement [/h2] Assume that the the offspring distribution is [itex]P(Y=y)=\left(\frac{1}{2}\right)^y\frac{1}{3}[/itex] [itex]y=0,1,2,\ldots[/itex] Show by induction that: [tex]G_n(s)=\frac{1-2^n-2(1-2^{n-1})s}{1-2^{n+1}-2(1-2^n)s}[/tex] [b]2. The attempt at a solution[/b] I can see that the distribution is geometric so: [tex]G(s)=\frac{p}{1-qs}=\frac{1}{3-2s}[/tex] I assume I have to show that: [tex]G_{n+1}(s)=\frac{1-2^n-2(1-2^{n-1})\frac{1}{3-2s}}{1-2^{n+1}-2(1-2^n)\frac{1}{3-2s}}[/tex] equals: [tex]\frac{1-2^{n+1}-2(1-2^{n})s}{1-2^{n+2}-2(1-2^{n+1})s}[/tex] The thing is, this seems like kind of a tedious question considering the amount of marks I'll get for it on my exam. Am I missing something here? Is there a "quick" way to do this? [/QUOTE]
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Calculus and Beyond Homework Help
Branching process, inductive proof
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