HW3

Due feb 7 2009 by 2:50pm. Please slide your HW under my office door in
Baker Hall 229E


* This is due tuesday feb 7
* We have the MT thursday feb 9 so the HW is short
* We will post solutions to HW3 right after class tuesday so you have time to 
check them before the MT


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(0) Don't get behind. Study the new material about branching processes and generating
carefully.

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(1) Consider a simple random walk starting at 0 in which each step is to the
right with probability p and to the left with probability q.
Let T be the number of steps until the walk first reaches b>0.
Show that E(T | T<infty) = b/|p-q|

There are various ways to prove this (generating functions, 1st step analysis, or other). 
However, I want to solve this using 1st step analysis because it is the easiest. Remember 
this is a conditional calculation.

The solution is very similar to the calculation in ex4 of the SRW with absorbing barriers.

You will need to solve a 2nd order difference equation. Hint to save you time: try a 
polynomial of degree 1 to find a particular solution to the inhomogeneous equation.
You also need 2 boundary conditions. One is trivial and the other require a bit of 
thought (similar to class notes).

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(2) Prove example 17(c) p 151 in the handout I gave today (it asks you to calculate
the generating function of the geometric distribution).
(short)

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(3) Consider a branching process whose family sizes have the geometric mass function 
f(k) = q p^k, k >=0, p+q=1, and let Z_n be the size of generation n.

Let T=min(n : Z_n=0) be the extinction time. Assume Z_0 = 1.
Find P(T=n).
For what values of p do we have E(T) < infinity?