Math 697 Fall 2014: Week 5
Exercise 1 (Card shuing) Suppose you have a deck of 52 cards. At each time you
shue the cards by picking a card at random and placing it on top of the deck. The state
space of this Markov chain consists of all permutation of the
Math 697 Fall 2014: Week 1
Exercise 1 For positive numbers a and b, the Pareto distribution with parameter a, b,
Pa,b has the p.d.f f (x) = aba xa1 for x b and f (x) = 0 for x < b. What is the inversion
method to generate Pa,b .
x
e
Exercise 2 The standar
Math 697 Fall 2014: Week 3
Exercise 1 (More algorithms to compute )
1. The estimator for constructed in class is based on the indicator RV
I = 1 if
V12 + V22 1
(1)
where V1 and V2 are uniform random variable on [1, 1]. As an alternative show that
you can
Math 697 Fall 2014: Week 2
Exercise 1 Consider a normal random variable Z with = 0 and = 1. Use Chernov
bounds to show that for any a > 0
2 /2
P (Z a) ea
Exercise 2 Consider a Poisson random variable N with mean . Use a Chernov bound
to show that for n >
Math 697 Fall 2014: Week 4
Exercise 1 (More on the Markov property) The Markov property means that the
future depends on the present but not on the past, i.e.,
P cfw_Xn = in | Xn1 = in1 , X0 = i0 = P cfw_Xn = in | Xn1 = in1 .
1. Show that the Markov pro
Math 697 Fall 2014: Week 8
Exercise 1 Consider the Markov chain on S = cfw_0, 1, 2, 3, with transition probabilities
P (0, 0) = 1 p0 , P (0, 1) = p0
P (j, j 1) = 1 pj ,
P (j, j + 1) = pj .
(a) Under which conditions on pj is the Markov chain positive rec
Math 697 Fall 2014: Week 9
Exercise 1 Given a branching process with the following ospring distributions determine
the extinction probability a.
(a) p(0) = .25, p(1) = .4, p(2) = .35
(b) p(0) = 5, p(1) = .1, p(3) = .4
(c) p(0) = .62, p(1) = .30, p(2) = .0
Math 697 Fall 2014: Week 7
Exercise 1 On a chessboard compute the expected number of plays it takes a knight,
starting in one of the four corners, to return to its initial position if we assume that at
each each play it is equally likely to make any of it
Math 697 Fall 2014: Week 1-solutions
b
x
Exercise 1 The cdf of the Pareto is F (x) = 1
is X = bU 1/a)
a
for x b. Thus the inversion method
Exercise 2 The cdf of the standardized logistic distribution has the p.d.f F (x) =
U
Thus the inversion method is X
Math 697 Fall 2014: Week 3
Exercise 1 (More algorithms to compute )
1. Two estimators:
Sn =
1
n
n
2
2
Ik , with I = 1cfw_U1 +U2 1 and E[Sn ] =
4
and var(Sn ) =
(1
4
1
n
) =
4
k=1
1
0.1685
n
n
Sn =
J = (1 U 2 )1/2 with E[Sn ] =
Jk ,
4
and var(Sn ) =
1
n
Math 697 Fall 2014: Week 2-Solutions
2 /2
Exercise 1 The MGF of Z is M (t) = et
. So the Chernov bound is
2
et /2
2
P (Z > a) min ta = et /2ta .
t0 e
2 /2
The minimum of t2 /2 ta is t = a and so we have P (Z > a) ea
.
t
Exercise 2 The MGF of N is M (t) =
Math 697 Fall 2014: Week 4
Exercise 1 (More on the Markov property)
1. Using the Markov property and Bayes formula
P cfw_X0 = i0 | X1 = i1 , Xn = in
P cfw_X0 = i0 , X1 = i1 , Xn = in
=
P cfw_X1 = i1 , Xn = in
P cfw_Xn = in | Xn1 = in1 P cfw_X1 = i1 |
Problem Set #2
Due: Thursday, 20 September 2012
Students registered in MATH 401 should submit solutions to three of the following
problems. Students in MATH 801 should submit solutions to all ve.
1. For a positive integer , the boolean lattice is the grap
Problem Set #3
Due: Thursday, 27 September 2012
Students registered in MATH 401 should submit solutions to three of the following
problems. Students in MATH 801 should submit solutions to all ve.
1
1. Let be a graph with at least two vertices, and let ()
Problem Set #5
Due: Thursday, 11 October 2012
Students registered in MATH 401 should submit solutions to three of the following
problems. Students in MATH 801 should submit solutions to all ve.
1. Let be a cycle in a connected weighted graph, and let be a
Problem Set #8
Due: Thursday, 1 November 2012
Students registered in MATH 401 should submit solutions to three of the following
problems. Students in MATH 801 should submit solutions to all ve.
1. Exhibit a perfect matching in the graph below or give a sh
Problem Set #4
Due: Thursday, 4 October 2012
Students registered in MATH 401 should submit solutions to three of the following
problems. Students in MATH 801 should submit solutions to all ve.
1. Let 1 , , be positive integers with 2. Prove that there exi
Problem Set #11
Due: Thursday, 22 November 2012
Students registered in MATH 401 should submit solutions to three of the following
problems. Students in MATH 801 should submit solutions to all ve.
1. (a) Calculate the chromatic polynomial of 1,3 by using t
Problem Set #12
Due: Thursday, 29 November 2012
Students registered in MATH 401 should submit solutions to three of the following
problems. Students in MATH 801 should submit solutions to all ve.
1. For the graph below, prove it is nonplanar or provide a
Non-homogeneous equations (Sect. 3.6).
We study: y + p (t ) y + q (t ) y = f (t ).
Method of variation of parameters.
Using the method in an example.
The proof of the variation of parameter method.
Using the method in another example.
Method of variation
Non-homogeneous equations (Sect. 3.5).
We study: y + a1 y + a0 y = b (t ).
Operator notation and preliminary results.
Summary of the undetermined coecients method.
Using the method in few examples.
The guessing solution table.
Operator notation and prelim
Autonomous systems (Sect. 2.5).
Denition and examples.
Qualitative analysis of the solutions.
Equilibrium solutions and stability.
Population growth equation.
Denition and examples
Denition
A rst order ODE on the unknown function y : R R is called
autonom
Modeling with rst order equations (Sect. 2.3).
Main example: Salt in a water tank.
The experimental device.
The main equations.
Analysis of the mathematical model.
Predictions for particular situations.
Salt in a water tank.
Problem: Describe the salt con
Exact equations (Sect. 2.6).
Exact dierential equations.
The Poincar Lemma.
e
Implicit solutions and the potential function.
Generalization: The integrating factor method.
Exact dierential equations.
Denition
Given an open rectangle R = (t1 , t2 ) (u1 , u
On linear and non-linear equations.(Sect. 2.4).
Review: Linear dierential equations.
Non-linear dierential equations.
Properties of solutions to non-linear ODE.
The Bernoulli equation.
Review: Linear dierential equations.
Theorem (Variable coecients)
Give
Review 2 for Exam 1.
5 or 6 problems.
No multiple choice questions.
No notes, no books, no calculators.
Problems similar to homeworks.
Exam covers:
Linear equations (2.1).
Separable equations (2.2).
Homogeneous equations (2.2).
Modeling using dierential e
Second order linear ODE (Sect. 3.1).
Second order linear dierential equations.
Superposition property.
Constant coecients equations.
The characteristic equation.
The main result.
Second order linear dierential equations.
Denition
Given functions a1 , a0 ,
Variable coecients second order linear ODE (Sect. 3.2).
Summary: The study the main properties of solutions to second
order, linear, variable coecients, ODE.
Review: Second order linear ODE.
Existence and uniqueness of solutions.
Linearly dependent and in
The integrating factor method (Sect. 2.1)
Overview of dierential equations.
Linear Ordinary Dierential Equations.
The integrating factor method.
Constant coecients.
The Initial Value Problem.
Variable coecients.
Read:
The direction eld. Example 2 in Secti