Write up your solution to the problems in LaTex, Word, etc., print it, and bring it to class when it is due. If you cannot make it to class, then email it to me before class on the day it is due. Simulation and other Matlab work can be turned in as is.
Use Matlab to simulate the following simple experiment: Break a stick (of length 1) into two pieces at random, i.e., the break point is at u ~ U(0,1). Take the longer piece and break it into two pieces at random. Let X be the length of the longer piece. Estimate E(X) and Var(X). Plot an estimate of the density and cumulative distribution function of X.
Problems 1.1, 1.2, 1.3 from the text
Problems 1.8 and 1.12 from the text. Use Matlab for your calculations when necessary.
Let X_n be a random walk on {0,1,2,...,10} where states 0 and 10 are absorbing, and P(i,i+1) = p, P(i,i-1) = 1-p.
a) Starting from state 5, find the expected number of visits to state 5 before being absorbed, as a function of p (numerically, using Matlab). HINT: Construct the 9x9 matrix "Q" with p as a variable. Then loop i = 1:99, and let p = i/100. Compute E(# visits to state 5) inside the loop. When the loop is done, plot E(# visits to state 5) as a function of p.
b) Do the same thing by simulation for p = .5 and see that the answers check.
HINT: when you are simulating a random walk, there is no need to construct the transition matrix. At each stage, simply "flip a coin" to see if you go up or down.
Problems 1.16, 1.17 from the text.
This is the problem we started in class. You have a coin with P(H) = p, and you flip the coin repeatedly until the pattern HTTH first appears. Find E(T) as a function of p, where T is the time it takes for the pattern to first appear. Write down the Markov chain you are using to solve the problem, but do the calculations with Matlab.
Note on problem 1.16: This apprears to be a hard problem! It might only require a simple but clever trick, but I haven't seen it yet.
One thing is (almost) for sure. You need to know that if you have a symmetric random walk starting at 0, then the probability you hit state -1 before
you hit state n is n/(n+1). First, show that this is true. Then try the problem for N=3 (easy) and N=4 (not too bad), If you have time, try N=5.
Maybe I'll have a full solution by class time on wednesday, but I can't promise!
problems 2.1, 2.2, 2.3, 2.5, 2.7
Here's one more problem for next week. Let X(t) be a Poisson process with rate 2 events per unit time.
(a) Find the distribution of X(10) and the distribution of X(20)-X(5).
(b) Find Cov(X(10), X(20)-X(5))
quick comment on problem 2.1 I didn't realize what a mess the follow-up question are! Try to answer the first question
(when is the chain positive recurrent, null recurrent, transient), and don't worry about the other parts until you've finished everything else.
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problems 3.1, 3.2, 3.3, 3.8
Midterm Exam: The midterm exam will consist of a take-home part and an in-class part. The in-class part will be Wednesday, 10/13.
I do not expect that part of the test to take up the entire class time. The questions will be relatively simple. The take-home part will
be more challenging, but no harder than some of the homework problems. It will be due (written up like a homework set) Wednesday 10/20.
Here's the first take home problem. You can start on it now if you want.
(1) For the "shoe-shine" model we discussed in class, find the expected time until the first customer is turned away.
(An arriving customer is turned away if the store is in state 2a or 2b.) You may want to simulate the model to check your answer,
but you are not required to. If you get the correct answer via simulation you will get partial credit,
even if your analytical/numerical solution is wrong. Use \lambda = 5, mu_1 = 10, \mu_2 = 20.
I took a look at all the in-class exams, and although nobody looked very happy leaving the room this afternoon, the results were actually very good. Everybody got an A or a B. Here is the take-home test due next Wednesday 10/20.
Homework problems due Wednesday, November 3.
Chapter 5, #1,2,4,6
Here are a couple more problems on martingales, due Wednesday, November 10.
Chapter 5, #7, 12
Don't be confused by the typo in #7. ( S(T) should be S_T )
Here's a fun problem to work on over the break: Chapter 8, problem 8.4.
Try to get an analytic solution in each case - no complicated integrals as answers! Parts (c) and (d) are tricky.
Here's the last homework assignment of the semester:
Chapter 8, #9,10,11
Next week we will briefly go over material on stochastic integration on Monday.
I will also post a take-home final exam before next Wednesday, December 8.
The exam will be due the following Wednesday, December 15.
Next Wednesday class will be a review of the material we covered this semester.
Here's the final exam .