# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

3 Pages

### Practice_Final_2008

Course: MATH 6397, Fall 2008
School: U. Houston
Rating:

Word Count: 421

#### Document Preview

Processes Stochastic - Spring 2008 Practice Problems for Final Exam Bernhard Bodmann, PGH 636 Duration: 150 minutes First Name: Last Name: Show all work. No points will be given for numerical answers without working being shown. (1) Consider the (continuous-time) Poisson process {Nt }t0 , which has independent increments on disjoint intervals, with distribution given by P(Nt Ns = k) = ((t s))k (ts) e k! for...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Texas >> U. Houston >> MATH 6397

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
Processes Stochastic - Spring 2008 Practice Problems for Final Exam Bernhard Bodmann, PGH 636 Duration: 150 minutes First Name: Last Name: Show all work. No points will be given for numerical answers without working being shown. (1) Consider the (continuous-time) Poisson process {Nt }t0 , which has independent increments on disjoint intervals, with distribution given by P(Nt Ns = k) = ((t s))k (ts) e k! for all s, t 0, k {0, 1, 2, . . .} and a xed parameter > 0. Show that the process Mt = Nt t is a martingale. (2) Suppose {Bt }t0 is a standard Brownian motion starting at B0 = 0. Let Bt = tB1 Bt . (a) Compute E[Bs Bt ]. Give a reason why B1 is independent of ({Bs : 0 s 1}). (b) Compare for xed s [0, 1] the distributions of Bs and B1s . 1 (3) Let Wt be a Brownian motion with drift parameter , that is Wt = Bt + t. (a) Show that for any real > 0 V (t) = eWt (+ 2 )t 2 is a martingale with respect to the ltration Ft = ({Ws : 0 s t}). (b) Taking = 2 in (a) you may conclude that V0 (t) = e2W (t) is a martingale. By using a stopping time argument or otherwise show that the probability that the Brownian motion with drift reaches b > 0 before a < 0 is 1 e2a . e2b (a) e2a (4) Let Bt denote standard Brownian motion and Ft the -algebra generated by the random variables {Bs }0st . Let Yt = max0st Bs . Use the reection principle to show that for all t 0, the distribution of Yt is identical to that of |Bt |. (b) Compute for xed t 0, the family of random variables {Sh }0<h<1 Sh = Bt+h Bt . h Show that this family has diverging norm in L2 (, F, P), that is, 0<s<1 2 sup E[Sh ] = . 2 (5) Suppose that {Xn } is a Markov chain with countable state space S = N and transition probability matrix P = (Pij ). Suppose (Vi ) is a right eigenvector for P with eigenvalue i.e. for all i N, Pij Vj = Vi j such that E[|VXn |] < for all n. Show that Yn = VXn n is a m...

Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

U. Houston - MATH - 6397
Stochastic Processes - Spring 2008Bernhard Bodmann, PGH 636 Exercise Sheet 2, with Solutions Do all Exercises individually. (1) Let X1 , X2 , . . . , Xn , . . . be i.i.d. with P(Xi = 1) = p and P(Xi = 1) = 1 p = q. Let a, b N. Dene Sn = X1 + . . .
U. Houston - MATH - 4377
Exam 2 Math 4377 November 20, Fall 2008Show all the work for full credit. In this exam no calculators are allowed. Answer all of the following questions. The maximum score is 150. 1. (15 Points) Let V = R2 and W be the vector space of all real 22 m
U. Houston - MATH - 6397
Stochastic Processes - Spring 2008Practice Problems for Final ExamBernhard Bodmann, PGH 636 Duration: 150 minutes First Name: Last Name:Show all work. No points will be given for numerical answers without working being shown.(1) Consider the (c
U. Houston - MATH - 4377
Department of MathematicsUniversity of HoustonMath 4377 Advanced Linear AlgebraFall 2008Homework Set 9, due Tuesday, Nov 4, 1pmSection 3.51 In R3 , let 1 = (1, 0, 1), 2 = (0, 1, -2) and 3 = (-1, -1, 0). (a) If f is a linear functional on R3
U. Houston - MATH - 4377
Department of MathematicsUniversity of HoustonMath 4377 Advanced Linear AlgebraFall 2008Homework Set 8, due Tuesday, Oct 28, 1pmSection 3.32 Let V be a vector space over the eld of complex numbers and suppose T is an isomorphism of V onto C3
U. Houston - MATH - 4377
Department of MathematicsUniversity of HoustonMath 4377 Advanced Linear AlgebraFall 2008Homework Set 10, due Tuesday, Nov 11, 1pmSection 3.61 Let n be a positive integer and F be a field. Let W be the subspace of all vectors (x1 , x2 , . . .
U. Houston - MATH - 4355
10 f f1 f2 f386420!2 !4!3!2!10123440 f f1 f2 f33020100!10!20!30!40 !4!3!2!1012341.75 1.7 1.65 1.6 1.55 1.5 1.45 1.4 1.35 1.3 0.2 a=0.3134 b=1.29490.30.40.50.60.70.80.91
U. Houston - MATH - 4377
U. Houston - MATH - 6397
U. Houston - MATH - 4377
U. Houston - MATH - 1432
Feb 7-9:30 AM1Feb 10-5:39 PM2Feb 10-5:45 PM3Feb 6-7:35 AM4Feb 6-7:36 AM5Feb 6-7:36 AM6Feb 6-7:37 AM7Feb 7-9:57 AM8Feb 7-9:57 AM9Feb 11-10:15 AM10Feb 11-10:20 AM11Jan 30 - 8:48 AM12Feb 10 - 9:29 AM13
U. Houston - MATH - 1432
123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960
U. Houston - MATH - 1432
Feb 20 - 8:59 AM1Feb 20 - 8:22 AM2Feb 20 - 8:30 AM3Feb 20-10:09 AM4Feb 20-10:16 AM5Feb 20-10:25 AM6Feb 15-8:37 AM7Feb 20 - 8:37 AM8Feb 15-8:34 AM9Feb 15-8:35 AM10Feb 15-8:35 AM11Feb 15-8:35 AM12Feb 15-8
U. Houston - MATH - 1432
1234567891011121314151617181920212223242526272829
U. Houston - MATH - 1432
Jan 30 - 8:48 AM1Feb 5-9:30 AM2Feb 5-9:30 AM3Feb 4-7:05 AM4Feb 4-7:07 AM5Feb 4-7:07 AM6Feb 4-7:08 AM7Feb 4-7:08 AM8Feb 8 - 9:38 AM9Feb 8 - 9:39 AM10Feb 8 - 9:39 AM11Feb 8 - 10:24 AM12Feb 9-8:54 AM13
U. Houston - MATH - 1432
Jan 30 - 9:59 AM1Jan 30-3:36 PM2Feb 4-8:59 AM3Jan 30-9:59 AM4Jan 30-9:58 AM5Jan 30 - 8:48 AM6Feb 1 - 9:43 AM7Feb 1 - 9:45 AM8Feb 1 - 9:47 AM9Feb 1 - 12:33 PM10Feb 3 - 8:54 AM11Feb 3 - 9:01 AM12Feb 3 - 9
U. Houston - MATH - 1432
Feb 22-8:03 PM1Feb 22-9:37 PM2Feb 17-6:11 PM3Feb 17-6:10 PM4Feb 23-10:14 AM5Feb 23-10:25 AM6Feb 18-6:58 AM7Feb 18-6:58 AM8Feb 19-9:06 AM9Feb 19-9:09 AM10Feb 19-9:09 AM11Feb 19-9:09 AM12Feb 19-9:09 AM1
U. Houston - MATH - 1432
Feb 17-10:36 PM1Feb 16-9:35 AM2Feb 13-8:00 AM3Feb 13-8:01 AM4Feb 13-8:02 AM5Feb 16-9:33 AM6Feb 16-9:34 AM7Feb 18-1:04 PM8Feb 18-1:05 PM9Feb 13-8:02 AM10Feb 20 - 8:59 AM11Feb 20 - 8:20 AM12Feb 20 - 8:32
U. Houston - MATH - 1432
12345678910111213141516171819202122
U. Houston - MATH - 3321
1234567891011
U. Houston - MATH - 1432
Jan 25-3:00 PM1Jan 23 - 9:02 AM2Jan 28-9:36 AM3Jan 28-9:37 AM4Jan 28-9:37 AM5Feb 1-9:18 PM6Feb 1-9:19 PM7Jan 28-9:38 AM8Feb 1-9:18 PM9Feb 1-9:20 PM10Jan 25-3:03 PM11Jan 28-9:38 AM12Feb 2-9:21 AM13
U. Houston - MATH - 3321
12345678910111213141516171819
U. Houston - V - 001
HOUSTON JOURNAL OF MATHEMATICS, Volume 1, No. 1, 1975.THE HOPF EXTENSIONTHEOREMFOR TOPOLOGICALSPACESKiiti MoritaIn celebration of the start of Houston Journal of Mathematics1. Introduction. Let X be a topological space. Then by the cover
U. Houston - MATH - 3334
Math 3334 Fall 2008 Professor William OttExamination 3Problem 1. Compute the integral2 0 4 2yex dx dy. Hint: Use the Fubini theorem. Problem 2. Prove that 4e5[1,3][2,4]2ex2+y 2dA4e25 .Hint: You do not need to evaluate the integral
U. Houston - MATH - 3334
Math 3334 Fall 2008 Professor William OttFinal exam information1. Topology and geometry of Rn 1.1. Concepts/denitions. (1) (2) (3) (4) (5) (6) (7) norm on Rn , triangle inequality, reverse triangle inequality inner (dot) product on Rn , orthogonal
U. Houston - MATH - 3334
Math 3334 Fall 2008 Professor William OttAssignment 2: due Wednesday, September 10, 2008Problem 1. We show that the notions of open set and closed set are dual to each other. (a) Prove that if A Rn is open, then Rn \ A is closed. (b) Prove that if
U. Houston - MATH - 3334
Math 3334 Fall 2008 Professor William OttAssignment 8: due Wednesday, December 10, 2008Problem 1. Marsden/Tromba: Chapter 6.4 (a) MT 6.4.1 (b) MT 6.4.6 (c) MT 6.4.9 (d) MT 6.4.17 Problem 2. (limitation of the Riemann integral) Let Q = {x R : x = p
U. Houston - M - 6325
Math 6325 Spring 2009 Professor William OttAssignment 2: due Thursday, February 19, 2009Problem 1. Let m N satisfy m 2. For x [0, 1), let (xi ) denote the base-m decimal expansion of x i=1 (assuming this expansion is unique). For 0 k m 1 and N
U. Houston - MATH - 3334
Math 3334 Fall 2008 Professor William OttAssignment 4: due Monday, October 20, 2008Problem 1. Marsden/Tromba problems (a) MT 3.2.6 (b) MT 3.3.10 (c) MT 3.3.11 (d) MT 3.3.19 (e) MT 3.3.34 (f ) MT 3.3.41 Problem 2. A function f : R R is analytic at
U. Houston - MATH - 3334
Math 3334 Fall 2008 Professor William OttAssignment 1: due Wednesday, September 3, 2008(1) Prove that for x , y Rn , we have x y x y . This inequality is known as the reverse triangle inequality. Hint: use the triangle inequality. (2) Problem 1.5
U. Houston - MATH - 3334
Math 3334 Fall 2008 Professor William OttCourse SyllabusOffice: PGH 603 Office hours: Tu 1011 AM, Th 12 PM Textbook: Vector Calculus (5th edition) by Marsden and Tromba Prerequisite: Math 3333 Overview. We will study multivariable calculus from a r
U. Houston - MATH - 3334
Math 3334 Fall 2008 Professor William OttAssignment 6: due Monday, November 17, 2008Problem 1. Marsden/Tromba: Section 5.2 (a) 5.2.4 (b) 5.2.11 (c) 5.2.12 Problem 2. Marsden/Tromba: Section 5.3 (a) 5.3.2a (b) 5.3.2d (c) 5.3.7 (d) 5.3.9 Problem 3. M
U. Houston - MATH - 3334
Math 3334 Fall 2008 Professor William OttAssignment 5: due Wednesday, October 29, 2008Problem 1. Marsden/Tromba: Section 3.4 (a) MT 3.4.11 (b) MT 3.4.12 (c) MT 3.4.15 (d) MT 3.4.20 Problem 2. Marsden/Tromba: Section 3.5 (a) MT 3.5.2 (b) MT 3.5.8 (c
U. Houston - MATH - 3334
Math 3334 Fall 2008 Professor William OttAssignment 3: due Wednesday, October 1, 2008Problem 1. Let A = (aij ) be an (m n)-matrix with aij R for all 1 we have a11 a12 a1n a21 a22 a2n . . . . . . . . . . . . am1 am2 amn Define 1
U. Houston - MATH - 3334
Math 3334 Fall 2008 Professor William OttExamination 2 information1. Material Exam 2 covers Chapter 3 of Marsden/Tromba. 1.1. Theoretical results. You should understand and be able to state the following theoretical results and use them to solve pr
Penn State - ZUX - 103
An Improved Approach to Passive Testing of FSM-based SystemsHasan Ural, Zhi Xu and Fan Zhang SITE, University of Ottawa, Ottawa, Ontario, K1N 6N5, Canada {ural, zxu061, fzhang}@site.uottawa.ca AbstractFault detection is a fundamental part of passiv
Penn State - IE - 327
IE 327 Introduction to Work DesignDr. Andris Freivalds Class #24IE 327 1Ch. 17 Wage Incentives Can workers be motivated? Blue-collar: White-collar: Direct financial plans = pay proportional to output Indirect plans: Fringe benefits Vacat
Penn State - GEOSC - 001
Geoscience 001 Spring 2006 Field Trip Lab The Bellefonte Formation and Nealmont Formations Materials: Students: hand lens, field notebooks Site: Hwy 322 By-Pass, 2 locations, along the east side of the road. This is a busy road, so stay well off the
U. Houston - FINA - 4355
U. Houston - FINC - 3331
ANSWERS TO END-OF-CHAPTER 6 QUESTIONS6-1a. The probability distribution for complete certainty is a vertical line. b. The probability distribution for total uncertainty is the X-axis from - to +.6-2Security A is less risky if held in a divers
U. Houston - COSC - 6365
Computer Science 6365February 26, 2008Lecture 13: VectorizationProfessor: S. Lennart Johnsson TA:Wei Ding1VectorizationVectorization is a set of techniques for making effective use of pipelined functional units, such as load/store units, a
U. Houston - COSC - 6365
Computer Science 6365February 7 and 12, 2008Lecture #8 and 9: Interconnection NetworksProfessor: S. Lennart Johnsson TA:Wei Ding1NetworksA bus based parallel architectures is practical only for a small number of processors. For several tho
U. Houston - COSC - 6365
Computer Science 6365March, 27 2008Lecture #20-21: Dense matrix multiplicationProfessor: S. Lennart Johnsson TA:Wei Ding1MatrixVector MultiplicationSeveral scenarios are possible depending upon whether or not the data is external or intern
U. Houston - COSC - 6365
Computer Science 6365April 17, 2008Lecture #26: Fast Fourier Transforms IProfessor: S. Lennart Johnsson TA: Wei Ding1The Fast Fourier TransformThe Fast Fourier Transform, the FFT, is one of the most widely used algorithms in science and e
Michigan State University - CSE - 838
Pipelined Merging of Two sorted list in a constant time (Coles Algorithm) Leaves contain the value Internal nodes merge at each time by updating the values Lv: the sequence of values of descendants of v Qv(j): At time j, a sorted sequence v has.
Penn State - MKTG - 485
Penn State - MATH - 110
Compound interest. If one invests P dollars at an annual interest rate of i percent then the return S1 at the end of the rst year will be S1 = P + rP or S1 = P (1 + r) where r = .01i. The return S2 at the end of the second year can be viewed as inve
Penn State - LAF - 243
Request For A New ApplicationDate Submitted: April 24, 2007 Submitted by: Lindsay Federoff Purpose: To develop an application that converts Kilometers into Miles. Application Title: Algorithms: Kilometers to Miles Converter Miles = Kilometers * 0.62
Penn State - LAF - 243
Request For A New ApplicationDate Submitted: April 19, 2007 Submitted by: Lindsay Federoff Purpose: To create an application for the Paint Dept. to convert liters to pints and gallons for McIntyres Hardware Store. Application Title: Algorithms: Metr
Penn State - EJS - 5116
Erich Schaefer 9/19/06 MIS 204 There are many advantages to using a computer in todays world. First and foremost would be the speed at which information can be conveyed. The list of things that a computer can process is virtually never ending. From m
Penn State - EJS - 5116
Request For A New ApplicationDate Submitted: 11/30/06 Submitted by: Erich Schaefer Purpose: To create a temperature converterApplication Title: Algorithms:Temperature Converter C= (F-32) * 5/9Notes:ApprovalsApproval Status: Approved By: Dat
Penn State - EJS - 5116
Request For A New ApplicationDate Submitted: November 11, 2006 Submitted by: Erich Schaefer Purpose: The Personnel department often is asked to do a quick computation of a customers state tax. Employees would save time and provide more accurate info
Penn State - EJS - 5116
Request For A New ApplicationDate Submitted: 12/5/06 Submitted by: Erich Schaefer Purpose: To convert kilograms into both pounds and ouncesApplication Title: Algorithms:Weight converterNotes:ApprovalsApproval Status: Approved By: Date: Assi
Penn State - GES - 5024
Rebecca Colabaugh Tamara Leone Gretel Sheasley Ed Wasko Shopping on the Web The web has become such a large place with so much to offer. The internets main intention was to share information, but now it is used for a vast variety of things. It has al
Penn State - JMG - 520
Jennifer M. Gerrardjmg520@psu.eduCurrent Address: 117 Jordan Hall State College, PA 16801OBJECTIVE EDUCATIONPermanent Address: 7622 Huntmaster Lane McLean, VA 22102__To obtain a position in the marketing field, special interest in research an
Penn State - MES - 121
COMMON LAYER 2 DEVICES AND FUNCTIONALITIES1Introduction7 6 5 4 3 2 1Routers, PAD's, X.25 switches Bridges, LAN switches, ATM switches and terminal servers Transceivers, repeaters, hubs, FDDI concentrators, MSAU's, modems, terminal adapters, DS
Penn State - CEB - 5111
Wireless Devices in HealthcareCourtney BellObjectives The use of Wireless devices Type of devices and software Uses in Nursing care Legal and Ethical issues Advantages and disadvantages for the NurseWireless Devices Wireless devices such a
Penn State - MIS - 204
Josh Yorio MIS204 2/6/06 1. Name and describe the various types of software in each of these categories, Business, Graphic &amp; Multimedia, Educational, &amp; Communication. Business software is application software that assists people in becoming more eff
U. Houston - MATH - 1313
LINEAR PROGRAMMING: A GEOMETRIC APPROACH33.1 3.2 3.3Graphing Systems of Linear Inequalities in Two Variables Linear Programming Problems Graphical Solution of Linear Programming ProblemsMany practical problems involve maximizing or minimizing
U. Houston - ELED - 4310
I decided to choose a child from my moms preschool class for the reading interest inventory. My mom chose Keely, one of the four year olds, for me to use because she is one of the smartest students in her class. While my mom was reading a book to the
U. Houston - QUEST - 4310
I decided to choose a child from my moms preschool class for the reading interest inventory. My mom chose Keely, one of the four year olds, for me to use because she is one of the smartest students in her class. While my mom was reading a book to the