Question

A koala sits in one of five trees. Every so often, it climbs down and moves to a different tree (chosen at random). This happens an average of once per hour.

a.) To analyze this process, it will be easier to consider the equivalent process where the koala moves to a uniformly random tree, potentially the same tree it is already in, at rate $\lambda$. What is $\lambda$ so that this process is the same as the original one?

b.) What is the probability that after two hours, the koala is in a different tree to its initial one?

Answer 1

We want to pick $\lambda$ so that the expected time to change trees is 1 hr

After lambda time, the Koala changes trees with probability 4/5

So the expected time is

$\sum_{k=1}^\infty k\lambda\left(\frac{1}{5} \right )^{k-1}\frac{4}{5}$

$=\frac{4}{5}\lambda \sum k(1/5)^{k-1}$

$\sum_{k=0}^\infty x^k = \frac{1}{1-x}$

DIfferentiate both sides

$\sum_{k=1}^\infty kx^{k-1}= \frac{1}{(1-x)^2}$

Plug in x=1/5

To get the required sum is 25/16

So expected time is

$\frac{4}{5}\times\lambda\times\frac{25}{16} = \frac{5}{4}\lambda = 1hr$

Lambda = 48mins.

Given lambda b) is easy to calculate