STOCHASTIC PROCESSES
EXERCISE SHEET 7
EXERCISE 1
Customers arrive in a bank with two counters at a rate t in any small time interval t.
Service at either counter is exponentially distributed with rate . What is the expected
length of the customers waiting
Exercise sheet 2
Probability of landing k heads:
p
Tossing a
coin
2m times
k
q=1-p
Exercise sheet 2
Treat the problem as a symmetric random walk
with return to the origin
Unbiased coin
0 if n is odd, and
if n is even
Unrestricted random walks
The i-th ste
SOLUTIONS STATISTICS STAT4024:
STOCHASTIC PROCESSES
PROBLEM 1
Parts (a-b) are bookwork and identical to last years exam paper. Part
(c) is bookwork . Parts (d i) and (d ii) are unseen new problems.
Part a)
Dene:
Ck Event that the gambler is eventually rui
SOLUTIONS STATISTICS STAT4024:
STOCHASTIC PROCESSES
PROBLEM 1
Parts (a-c) are bookwork. Parts (d-e) are unseen problems similar to
tutorial examples.
a)
Dene:
Ck Event that the gambler is eventually ruined
when starting with initial capital k
uk
P (Ck )
D
Stochas(c Processes
Class Test 2
Solu(ons
Problem 1-a-i
[0 1 0] is a le8 eigenvector with eigenvalue 1
[0 1 0] is a sta(onary distribu(on
3 marks
Problem 1-a-ii
8 marks
The transi(on matrix has only one eige
Revision lecture:
Random walks
Lectures on
Stochastic Processes
Dirk Husmeier
Introduction to random walks
The set of all
possible states is
known as the
state space
.
.
Introduction to random walks
The ith step taken:
Position after n steps:
.
.
Introduc
SOLUTIONS: STOCHASTIC PROCESSES
CLASS TEST 15 NOVEMBER 2013
PROBLEM 1
a)
Dene:
Ck Event that the gambler is eventually ruined
when starting with initial capital k
uk
P (Ck )
D
Event that gambler wins the rst play
D
Event that gambler loses the rst play
p
December 2013
XXX
EXAMINATION FOR THE DEGREES OF M.A., M.SCI. AND B.SC.
(SCIENCE)
Statistics STATS 4024
Stochastic Processes
Hand calculators with simple basic functions (log, exp, square root, etc.) may be used
in examinations. No calculator which can st
May 2013
10 May 2013, 9:30 -11:00 am
EXAMINATION FOR THE DEGREES OF M.A., M.SCI. AND B.SC.
(SCIENCE)
Statistics STATS 4024
Stochastic Processes
Hand calculators with simple basic functions (log, exp, square root, etc.) may be used
in examinations. No calc
1 December 2014
9:00-10:00 am
CLASS TEST AS PART OF THE
EXAMINATION FOR THE DEGREES OF
M.A., M.SCI. AND B.SC.
Statistics STATS 4024
Stochastic Processes
Hand calculators with simple basic functions (log, exp, square root, etc.) may be used in
examinations
15 November 2013
2:00-4:00 pm
CLASS TEST AS PART OF THE
EXAMINATION FOR THE DEGREES OF M.A., M.SCI. AND B.SC.
(SCIENCE)
Statistics STATS 4024
Stochastic Processes
Hand calculators with simple basic functions (log, exp, square root, etc.) may be used
in ex
Wednesday, 20th May 2015
2.00 pm - 3.30 pm
EXAMINATION FOR THE DEGREES OF M.Sci., M.Sc.
and M.Res.
Stochastic Processes - Level M
Hand calculators with simple basic functionsexp,square
(log,
root,
etc.) may be used
in examinations.
No calculator which can
Exercise sheet 7
Solutions
r=2
Summary: multiple servers
r=2
The operation theatres
must operate at a rate of
/1.7
= 0.6
Recall: two equivalent servers
Probability that a
counter becomes free
if exactly one counter
is busy
Probability that a counter
beco
Exercise sheet 5
Solutions
Summary:
Differential equations for a
memory-free discrete process
Do these differential equations dene the
Poisson process?
Do these differential equations dene the
Poisson process?
We have shown:
Do these differential equation
STOCHASTIC PROCESSES
EXERCISE SHEET 1
EXERCISE 1
In the standard gamblers ruin problem with total stake a and gamblers stake k and
the gamblers probability of winning at each play p, calculate the probability of ruin
and the expected duration of the game
Solu%ons
Stochas%c Processes
M-level assignment
2014/15
Solu%on taken from C. Bishop PaCern Recogni%on and Machine
Learning, Springer Verlag 2006, Chapter 11
Metropolis-Has0gs algorithm:
Solu%on taken from C. Bishop Pa
STOCHASTIC PROCESSES
EXERCISE SHEET 5
EXERCISE 1
Consider a discrete stochastic process in continuous time, where pn (t) denotes the probability
that n Z+ discrete events have occurred in the time interval [0, t], t > 0. No event has
0
occurred at time 0:
STOCHASTIC PROCESSES
EXERCISE SHEET FOR M-LEVEL STUDENTS
Read the articles Understanding the Metropolis-Hastings Algorithm, by Chibb and Greenberg, and
Introducing Markov chain Monte Carlo, by Gilks, Richardson and Spiegelhalter. Then answer the
following
STOCHASTIC PROCESSES
EXERCISE SHEET 3
EXERCISE 1
Show that for a stochastic matrix T , and any row vector x, we have |xT | |x|,
where |x| = i |xi |. Deduce that all eigenvalues, , of T must satisfy | 1.
EXERCISE 2
Show that a stochastic matrix always has
STOCHASTIC PROCESSES
EXERCISE SHEET 2
EXERCISE 1
A coin is spun 2n times and the sequence of heads and tails is recorded. What is the
probability that the number of heads equals the number of tails after 2n spins?
EXERCISE 2
Two walkers each perform symme
STOCHASTIC PROCESSES
EXERCISE SHEET 6
EXERCISE 1
Whilst in use, an oce printer is observed to have a failure rate function (or hazard function) of
r(t) = 2t per hour where = 0.00028 hours2 . In other words, r(t) is the probability that the
printer fails w
STOCHASTIC PROCESSES
EXERCISE SHEET 4
EXERCISE 1
Show that if x1 , x2 , . . . , xm is a homogeneous Markov chain, then the reverse chain xm , xm1 , . . . , x1
is also a Markov chain, but not necessarily homogeneous. Which additional condition is
needed to
Exercise sheet 4
Solutions
Use the fact that the forward
chain is a Markov process
Bayes rule
then
does not depend
on k and is,
therefore,
homogeneous
Eigenvalues:
Only one eigenvalue
with modulus 1
the stationary distribution is the
unique limiting distr
Solutions
Exercise sheet 6
Lectures on
Stochastic
Processes
Dirk Husmeier
Recall: Properties of the hazard function
time
No breakdown until
Device does not deteriorate with age !
Three components in parallel have times to failure
which are independent and
Exercise sheet 1
Solutions
Probability that gambler A is eventually
ruined when starting with initial capital k
Expected duration of the game
Without
a draw
With
a draw
With
a draw
Same boundary
conditions
With
a draw
Homogeneous difference equations
Same
University
HofGlasgow
Thursday, 3rd May 2012
09.30 am - 11.00 am
VIA VERI'I'AS VITA
EXAMINATION FOR THE DEGREES OF lVI.A., M.SCI. AND B.SC.
(SCIENCE)
Statistics e Stochastic Processes
Hand calculators with, simple basic functions (log, esp, square root,