Homework 4 (Stats 620, Fall 2013)
Due Tuesday Oct 8, in class
1. Let A(t) and Y (t) denote respectively the age and excess at t. Find:
(a) Pcfw_Y (t) > x|A(t) = s.
(b) Pcfw_Y (t) > x|A(t + x/2) = s.
(
Homework 5 (Stats 620, Fall 2013)
Due Tuesday Oct 22, in class
1. Prove that if the number of state is n, and if state j is accessible from state i, then it is
accessible in n or fewer steps.
k
Soluti
Bernardo DAuria
Stochastic Processes - 2010/11
Notes
March 01st , 2011
Exercise 2.10 pag 90 in [1]
Buses arrive at a certain stop according to a Poisson process with rate . If you take the bus from th
Homework 2 (Stats 620, Fall 2013)
Due *Tuesday Sept 24*, in class
1. Let cfw_N (t), t 0 be a Poisson process with rate . Calculate E N (t)N (t + s) .
Comment: Please state carefully where you make use
IEOR 6711: Stochastic Models I, Professor Whitt
Solutions to Homework Assignment 6
Problem 3.1 (a) Yes since it is a contrapositive statement of (3.2.1).
(b) No. Note that we can have Xn+1 = 0, so tha
Homework 7 (Stats 620, Winter 2015)
Due Tuesday March 24, in class
1. Show that a continuous-time Markov chain is regular, given (a) that i < M < for all i or
(b) that the corresponding embedded discr
Homework 3 (Stats 620, Fall 2013)
Due Tuesday October 1, in class
1. Prove the renewal equation
t
m(t x) dF (x)
m(t) = F (t) +
0
Hint: One approach is to use the identity E[X ] = E E[X |Y ] for approp
Homework 1 (Stat 620, Fall 2013)
Due Thu Sept 12, in class
1. (a) Let N denote a nonnegative integer-valued random variable. Show that
Pcfw_N k =
E[N ] =
k=1
Pcfw_N > k .
k=0
(b) In general show that
Statistics 620
Final exam, Fall 2013
1. Suppose that traffic on a road follows a Poisson process with rate cars per minute. A chicken
needs a gap of length at least c minutes in the traffic to cross t
Homework 1 (Stat 620, Fall 2013)
Due Thu Sept 12, in class
1. (a) Let N denote a nonnegative integer-valued random variable. Show that
Pcfw_N k =
E[N ] =
k=1
Pcfw_N > k .
k=0
(b) In general show that
Homework 4 (Stats 620, Fall 2013)
Due Tuesday Oct 8, in class
1. Let A(t) and Y (t) denote respectively the age and excess at t. Find:
(a) Pcfw_Y (t) > x|A(t) = s.
(b) Pcfw_Y (t) > x|A(t + x/2) = s.
(
Homework 2 (Stats 620, Fall 2013)
Due *Tuesday Sept 24*, in class
1. Let cfw_N (t), t 0 be a Poisson process with rate . Calculate E N (t)N (t + s) .
Comment: Please state carefully where you make use
Homework 7. Due by 5pm on Wednesday 10/16.
Collaborative research & Human participants and animal subjects
Most statisticians do not have to worry directly about running large scientic research groups
Homework 9 (Stats 620, Fall 2013)
Due Tuesday Dec 3, in class
1. Consider successive ips of a coin having probability p of landing heads. Use a martingale
argument to compute the expected number of ip
4. Markov Chains
A discrete time process cfw_Xn , n = 0, 1, 2, . . .
with discrete state space Xn cfw_0, 1, 2, . . . is a
Markov chain if it has the Markov property:
P[Xn+1 = j |Xn = i, Xn1 = in1 , .
Stats 620 Final Exam
The exam will be in the usual classroom (271
Dennison), on Thursday April 23, 1:30-3:30.
Four questions will cover the material since the
midterm, and one question will be based
Statistics 620
Midterm exam, Fall 2012
Name:
UMID #:
Midterm Exam
There are 4 questions, each worth 10 points.
You are allowed a single-sided sheet of notes.
You are not allowed to make use of a ca
Statistics 620
Final exam, Fall 2012
1. This question asks you to confirm the validity of a variation on the Hastings-Metropolis algorithm. Let Q = cfw_qij be a symmetric irreducible transition proba
The Duality Principle for Random Walks
(This topic will not be in the final exam)
Pn
For Sn = i=1 Xi with X1 , X2 , . . . iid, we note
that (X1 , . . . , Xn ) has the same joint distribution
as (Xn ,
3. Renewal Theory
Definition 3 of the Poisson process can be
generalized: Let X1 , X2 , . . . , iid F (x) be
non-negative interarrival times. Set
Pn
Sn = i=1 Xi and N (t) = max cfw_n : Sn t.
Then cfw
Stats 620: Applied Probability
and Stochastic Models
1. Review of Probability
An event E is a subset of a sample space S.
The probability of E is written as P(E).
Formally, P(E) may be undefined fo
Statistics 620
Final exam, Fall 2013
Name:
UMID #:
There are 5 questions, each worth 10 points.
You are allowed two single-sided sheets of notes.
You are not allowed to make use of a calculator, or
Statistics 620
Final exam, Fall 2012
Name:
UMID #:
There are 5 questions, each worth 10 points.
You are allowed two single-sided sheets of notes.
You are not allowed to make use of a calculator, or
7. Brownian Motion & Diffusion Processes
A continuous time stochastic process with
(almost surely) continuous sample paths which
has the Markov property is called a diffusion.
almost surely means wi
Stats 620 Winter 2015 Midterm exam
The exam will be in class, on Thursday
February 26. You are allowed a single-sided page
of notes You are not allowed a calculator or
any other electronic device. Th
Homework 9 (Stats 620, Winter 2015)
Due Tuesday April 7, in class
1. Consider successive flips of a coin having probability p of landing heads. Use a martingale
argument to compute the expected number
Statistics 620
Final exam, Fall 2010
1. Let cfw_Sn , n = 0, 1, 2, . . . be a random walk, i.e., S0 = 0 and Sk = k=1 Xj for k > 0 with
j
cfw_Xn , n = 0, 1, 2, . . . being a sequence of independent, i
6. Martingales
For casino gamblers, a martingale is a betting
strategy where (at even odds) the stake doubled
each time the player loses. Players follow this
strategy because, since they will eventua
4. Markov Chains
A discrete time process cfw_Xn , n = 0, 1, 2, . . .
with discrete state space Xn cfw_0, 1, 2, . . . is a
Markov chain if it has the Markov property:
P[Xn+1 = j|Xn = i, Xn1 = in1 , .