- #1
spitz
- 60
- 0
Homework Statement
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]
2. The attempt at a solution
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?