Solve a Branching Process: Xn & F(s) - Get Help Now!

[Tranquillity]

I attach the following question about branching processes. Xn is the size of the nth generation. F(s) is the pgf of Z, the offspring distribution.

Any hints/help on how to proceed with my proof would be greatly appreciated!

Regards

 

[Jameson]

Your attachment shows your attempt which is great because people don't usually help without seeing effort. Might I suggest though trying to retype your attempt in a new post using Latex? You can take a look at http://www.mathhelpboards.com/showthread.php?27-How-to-use-LaTeX-on-this-site to see how to use Latex on MHB.

Jameson

 

[Tranquillity]

The suggestion is good, though this question involves a proof and a lot of notation is involved, I would prefer to get hints and continue uploading my handwritten attempts, to show every time exactly what I think and where I have a problem.

Are you able to help me with the exact question? Thanks!

Regards

 

[Moo1]

Hello,

A bit late, but I can answer. Don't condition by $$X_1$$, but by $$X_{n-1}$$. Then note that $$S_n=\sum_{i=1}^{X_{n-1}} Z_i^{(n)}$$ to write the expectation of a product : $$E\left[\prod_{i=1}^{X_{n-1}} s^{Z_i^{(n)}}\bigg|X_{n-1}\right]$$ and finish it off.

 

[Tranquillity]

Hello, thanks for the reply! I have figured it out! I was asked by the exercise to condition on X1. I have finished my proof which conditions on X1. Thanks again :)

 

[Moo1]

Well it'd be nice if you shared it with us, for some people may need to have this kind of proof at hand :) (although I personally don't need it (Evilgrin))

 

[Tranquillity]

Well it'd be nice if you shared it with us, for some people may need to have this kind of proof at hand :) (although I personally don't need it (Evilgrin))

Try this :) It conditions on something similar that was so near to what I was trying to do, If you undestand what I am posting you can derive what I am asked to! Any more help I could provide that! Regards!