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 k*th* 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 5*th* 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?

Related post:

Kudos to you if you find the correct solutions!

May 26th, 2010 at 12:11 pm

Nice questions! I also think that I had more fun writing my question than my students had solving it 🙂 Nevertheless, it’s always nice to see the smiles and giggles during the exam.

May 26th, 2010 at 12:35 pm

I’m not sure about the reproductive behavior of werewolves (I thought they were heterosexual, which might require a more detailed model than intended), but I’m pretty sure I could argue the premise of your second question. Even before Episode IV, it seems to me that the Rebel Alliance was introduce Storm Troopers to an absorbing state missing from your Markov chain.

May 26th, 2010 at 1:01 pm

i would assume that, as with vampires, “reproduction” refers to conversion from existing humans…

May 26th, 2010 at 1:40 pm

Paul and Aaron, I will admit that the mode of werewolf reproduction was not intended to be realistic, since I wanted to students to use a model that we covered in class. But to be honest, I know even less about werewolf lore than I do about vampire lore.

Tallys, the exam was a take-home exam, so I did not have the pleasure of smiles and giggles. I definitely regret that. It sounds like you handled things the right way!

May 26th, 2010 at 3:22 pm

I have no clue what she is talking about…but I love sexy smart chicks! I wish I could take you to a toxic narcotic show 🙂

May 30th, 2010 at 1:29 am

+1 for new and interesting set of questions.

Probably with the “new era” in mind? I was tired of “static” GRE/SAT questions like “so what would be the age of the daughter of father’s son if the mom’s age is twice as the cat’s age?…’

One more question to be added:

“Transformers-2” type actual intelligence (0:

June 1st, 2010 at 4:43 am

I am pleased to see that other people set exam questions which make students smile. One of my favourite Markov chain problems concerned the student who cycled between home and campus. The weather followed a two-state Markov process, states not-fine and fine when the student wanted to travel. (So you need two transition probabilities, p(fine-fine), p(not-not)) The student had N bicycles, initially all at home. He/she would ride if the weather was fine and there was a bicycle available at the start. (so the student could cycle from home and leave the bicycle there because the weather had changed. Or vice versa.) Bicycles were stored wherever they were left, so in time the N bicycles were divided between home and campus. For varying N, find the steady–state probability that the student has a bicycle available when the weather is fine.