I’m recovering from the end of the semester. I’m looking forward to a return to regular blogging. I’ll start writing about the end of the semester. I decided to have some fun with my stochastic processes final this semester and to write questions about werewolves and Star Wars (I was inspired by Tallys Yunes’s vampire network flow). I’ll be honest, I think that I had more fun with this than did my students.

The werewolf question: The werewolf population in the Richmond area can be modeled as a linear growth birth and death process. Each werewolf independently reproduces at a rate of lambda = 0.15 werewolves/year and is killed by vampires at a rate of mu = 0.1/year. If the population started with a pack of three werewolves in the year 1860, find the average size of the werewolf population today (150 years later).

The Star Wars question (pre-Episode IV): Suppose that every month, Darth Vader organizes a gathering on the Death Star to build morale and promote bonding among the Storm Troopers. The Storm Troopers’ attendance at the gatherings is represented by a Markov chain. Given that a Storm Trooper has attended the last gathering (state 0), they go to the next gathering with probability p 0 . In general, given that they last attended the kth prior gathering, they go to the next gathering with probability p k, with 0 < p k < 1 , k = 0,1,2,3,4. Storm Troopers are required to attend a gathering every six months, and hence, given that they last attended the 5th prior gathering, they go to the next gathering with probability 1 (p 5 = 1) .

a. Define the Markov chain for this problem, specify the classes, and determine whether they are recurrent or transient.

b. What is the cumulative density function representing the number of months until a Storm Trooper first returns to the gathering (i.e., the first return to state 0)? Assume that they have just attended a gathering (i.e., they start in state 0).

b. In the long run, what is the proportion of Storm Troopers that have attended one of the last three gatherings?

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Kudos to you if you find the correct solutions!