STAY-HOME FINAL EXAM
PSTAT 160B – Winter 2021
Jean-Pierre Fouque
Instructions:
• Please submit your work as a pdf-file via Gradescope. Don’t forget to match your pages
with the problems.
• Submission deadline: March 19, 2021, at noon (12 p.m. (PST)).
• Academic Integrity at UCSB: “Papers, examinations, laboratory reports, and homework
must always be your own work. Cheating, plagiarism, and collusion are all forms of academic
dishonesty and are violations of the Student Conduct Code. Students found responsible for
these violations face possible suspension or dismissal from the University.”
• Academic Conduct at UCSB: “It is expected that students attending the University of
California understand and subscribe to the ideal of academic integrity, and are willing to bear
individual responsibility for their work. Any work (written or otherwise) submitted to fulfill
an academic requirement must represent a students original work. Any act of academic
dishonesty, such as cheating or plagiarism, will subject a person to University disciplinary
action. Cheating includes, but is not limited to, looking at another students examination,
referring to unauthorized notes during an exam, providing answers, having another person
take an exam for you, etc.”
• This is an open notes exam in the sense that you are welcome to refer to our lecture videos,
lecture notes (Dobrow’s book), and assignment sheets, but nothing else!
• This is not a collaborative exam. You are expected to take this exam in isolation with
no communication with other people. Communicating with others about the exam
problems is considered cheating.
• You must show all of your work and carefully justify your answers, otherwise points
will be deducted. You may justify your answers by either writing brief sentences explaining
your reasoning or annotating your math work with brief explanations. Explicitly cite re-
sults from class when you use them and briefly explain why they are applicable.
Introduce all processes, random variables etc. you use.
• Use a calculator to compute numerical values as final answers (if applicable).
Good Luck!
PSTAT 160B STAY-HOME FINAL EXAM
1. (10 points) Let pNtqtě0 be a Poisson process with parameter λ ą 0 and denote by S1, S2, . . .
the corresponding arrival times. Moreover, let f : RÑ R be a bounded function.
Show that
E
«
Ntÿ
i“1
fpSiq
ff
“
ż t
0
λ ¨ fpxq dx pt ě 0q.
2. (10 points) Claims are reported to a car insurance company according to a Poisson process
with an average of 6 claims per day. The USD amount of each claim follows an exponential
distribution with mean $2,000. Suppose that all claims and the total number of reported claims
are independent of each other.
(a) Compute the expected total sum of claims (i.e., the total USD amount) after one week.
(b) Find the probability that more than one claim with claim size $10,000 or higher will be
reported to the insurance company within one week.
3. (10 points) A three-state continuous-time Markov chain with state space S “ ta, b, cu has
distinct holding time parameters qa “ 3, qb “ 1, and qc “ 2. From each state, the process is
equally likely to transition to the other two states.
(a) Exhibit (i) the transition probability matrix for the embedded Markov chain, (ii) the
generator matrix, and (iii) the transition rate graph.
(b) Find the stationary distribution of the continuous-time Markov chain.
(c) Find the stationary distribution of the embedded chain.
4. (10 points) Two dogs – Lisa and Cooper – share a population of N P N fleas. Fleas jump from
one dog to another independently at rate λ per minute. Let Xt denote the number of fleas on
Lisa at time t (measured in minutes). We assume that pXtqtě0 is a birth-and-death process.
Suppose there are x P t0, 1, . . . , Nu fleas on Lisa at time t “ 0.
(a) Compute the expected number mxptq of fleas on Lisa at time t ą 0, i.e., find
mxptq “ ErXt |X0 “ xs.
Hint: Use Kolmogorov’s forward equation to show that the function mxptq satisfies the
linear ODE m1xptq “ ´2λmxptq `Nλ with mxp0q “ x. Then, recall that the solution to a
linear ODE of the form
f 1pxq “ a ¨ fpxq ` b, fp0q “ c
with constants a, b, c P R is given by
fpxq “
ˆ
c` b
a
˙
¨ eax ´ b
a
.
(b) Compute the limit limtÑ8 ErXt |X0 “ xs, i.e., the expected number of fleas on Lisa in
the long-run.
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PSTAT 160B STAY-HOME FINAL EXAM
5. (10 points) It is time for students to register for classes, and a line is forming at the registrar’s
office for those who need assistance. It takes the registrar an exponentially distributed amount
of time to service each student, on average 3 minutes. Students arrive at the office and get
in line according to a Poisson process at the rate of one student every 5 minutes. The arrival
times of the students are independent of the registrar’s service time.
(a) What is the long-term probability that there are not more than 3 students at the registrar’s
office?
(b) How long, on average in the long run, is a student at the registar’s office?
(c) What is the long-term expected number of students at the registar’s office waiting to be
served by the registrar?
6. (10 points) Let pBtqtě0 be a standard Brownian motion. We define the stochastic process
pXtqtě0 via
Xt “ e´tBet pt ě 0q.
(a) Show that pXtqtě0 is a Gaussian process.
(b) Find the mean and covariance functions.
7. (10 points) Let f, g : R Ñ p0,8q be two strictly positive probability density functions. More-
over, let X0, X1, X2, . . . be a sequence of i.i.d. random variables with probability density g.
Define
Y0 “ 1, Yn “
nź
i“1
fpXiq
gpXiq for all n ě 1.
Show that pYnqn“0,1,2,... is a martingale with respect to pXnqn“0,1,2,....
8. (10 points) Let pBtqtě0 denote a standard Brownian motion. Show that the stochastic differ-
ential equation
dXt “ ´ Xt
1` t dt`
1
1` t dBt, X0 “ 0,
admits the solution
Xt “ Bt
1` t , t ě 0.
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