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491 STAT/MATH Solutions to homework 2. 5.4.1 E(Z m Zn) = . Now given that Zm = k, Zn can be thought of as k independent chains of length m n. Thus E(Z n Z m = = = so E(Z m Z n ) = ( ) = . Hence Cov(Z m , Z n ) = = + = ( ) = Cov(Z n , Z m ) mm - 1 mm - 1 n - m VarZ m n - m - ( n - m )/2 ( n - m )/2 =m =m m =m so (Z m , Z n ) = VarZ n mn - 1 mn - 1 VarZ mVarZ n using Lemma 2 on p. 172. (n) (n) 6.2.2 Since pis > for n = n(i), and psi = for all n, i does not intercommunicate with s. Hence i and s are not in the same equivalence class. By Chapman-Kolmogorov (n pii ) = = = < , so i is a transient state. 6.3.1 Assume r<1. The chain jumps to a randomly selected state whenever it hits 0. Then it moves down or stays one step at the time until it hits 0 again. Hence all states intercommunicate, and the chain is persistent. Assuming a0 > 0 the period is 1. The mean recurrence time for 0 is clearly one (the step from 0 to a state j) plus the expected time to leave state j (1/1-r) times the expected value of the distribution (a0,a1,...) or 1+ . For other states than one the calculation is similar, except that one must separate out excursions for values lower than i (where i is not going to be reached). 6.3.3 (a) Assume 0<p<1/2. Since all states communicate they are all persistent. Since the diagonal elements are nonzero the chain is aperiodic. To find the mean recurrence time, 0 (1 - 2 p) + p 1 p = p 0 solve for the stationary distribution: p 0 2 p + p 1 (1 - 2 p) + (1 - p 0 - p 1 )2 p = p 1 whence = (1/4,1/2,1/4). Thus, using theorem 3 on p. 227, the mean recurrence time for states 0 and 2 are each 4, while that for state 1 is 2. I do not find it easy to compute the nstep transition probabilities. Here is one approach: in order to go from 0 to 0 in n steps we can either go from 0 to 0 n times, or we can go from 0 to 1, then go from 1 to 1 in n-2 steps, and then go from 1 to 0. Since we cannot go directly from 0 to 2 we do not need to worry about that route. Thus the diagonal elements of Pn satisfy (n p00 ) = + = + + = + We can solve these equations by substituting the first and third into the second, which then will be a difference equation (see handout on web site; the characteristic polynomial has a pair of complex conjugate roots which are not covered in the handout). Another approach is to represent P = B BT where is diagonal and B orthogonal. Then it is easy to power up P. (b) Assume 0<p<1/2. Again all states communicate, so the chain is persistent. Since the diagonal elements are all zero the chain is periodic. The period is 2 (since you can get from 0 to 0 by going 0 1 0 and the same for all the other states). The stationary distribution satisfies the following system of equations: 1 p + p 3 (1 - p) = p 0 p 0 (1 - p) + p 2 p = p 1 p 1 (1 - p) + p 3 p = p 2 p 0 p + p 2 (1 - p) = p 3 p 0 + p1 + p 2 + p 3 = 1 Adding the first and third equations we get 1 + p 3 = p 0 + p 2 and using the fifth we see that 1 + p 3 = p 0 + p 2 = 1 . Adding the first second and and using 2 = 1 - p 0 , p 3 = 1 - p 1 2 2 2 1 we get that 0 p + p 1 (1 - p) = 4 . Finally adding the first and the fourth and using all the previous relations we get 0 = 1 and thus = ( 1 , 1 , 1 , 1 ) . Hence the mean recurrence 4 4444 (n) time for each of the four states is 4. The calculation of pij can be done using matrix multiplication, but there does not seem to be a simple pattern, except that every other matrix has the pattern of the original matrix (with entries f(p) and 1 f(p) where f is a polynomial of degree n in p) and every other have nonzero terms replaced by zeros, and zero terms replaced by nonzero terms. 6.4.3 Since X n = + , truncated at 0 and K, with Yn independent of Xn-1 it is clear that the process is a Markov chain. The state space is {0,...,K} and pij = = + , for i > 0 and j < K, with p0 j = = , p0, K = and p j,K = + for j 1. In other words, the transition matrix is P= where Pl = . Solving for the stationary distribution we get the first equation o p0 + p 1 p0 = p 0 p0 1 - p0 P = p 0 1 . The second equation from which we get 0 + p 1 = or 1 = p 0 p0 p0 p0 becomes ( + + = = Using the expressions above we get 2 = p 0 fashion we get k = p 0 Pk k . Using that the stationary distribution must sum to one we see po Pk 1 k 0 = K p0 =K P p = that pkk and k Pi . In the geometric case we get 0 and k =0 0 i i = 0 p0 (1 - a )a k Pk = = . Hence k = is a truncated geometric distribution 1 - a K +1 = 1 - p where = (assuming p>1/2). p If one interprets the description of the process differently, the state space becomes {0,...,K-1}, and p00 = p0 + p1. A similar computation still works, but not quite as neatly. 1 - p0 - p1 P = p 0 2 . Continuing in the same 2 2 p0 p0 6.4.8 At each time a surviving particle either dies (with probability p) or survives (with probability 1-p), regardless of how long it has been in the chamber, and of how many other particles there are (this is another way of saying that the geometric distribution is memoryless). Hence, given that Xn-1 = k, the number surviving until time n is Bin(n,1-p). Independently, a Po( )-number of particles enter at time n. Hence the conditional distribution of Xn, given that Xn-1 = k, has pgf e + . Since i pij = p j i we multiply both sides by sj and sum to get G = = = = + = + Differentiating both sides we get Gp (s) = l el ( s - 1)Gp ( p + (1 - p)s) + el (s - 1) (1 - p)Gp ( p + (1 - p)s) l Letting s we get Gp (1) = l + (1 - p)Gp (1) so Gp (1) = ip i = . This is the expected p value of the stationary distribution. It seems reasonable to guess that the stationary distribution is Poisson (since a binomial sample from a Poisson is Poisson). Attempting G = e the right-hand side of the equation above yields = + , the desired result.
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CA.trend.coef.matrix.txt
Path: Washington >> STAT >> 498 Fall, 2008
Description: 060370002,2.93443293273918,-6.60175212411867,1.46086781027834 060371002,3.03280812436641,-6.44671558582038,-1.66919019159562 060371103,2.99947555742709,-6.23861259473043,-1.0092641065844 060371201,2.79750673204645,-6.36486904517982,-0.838880577044036...
CA.lcovar.subset.txt
Path: Washington >> STAT >> 498 Fall, 2008
Description: ID2,LAT,LONG,LANDUSE,PopDens,m_to_a1,m_to_a2,m_to_a3,m_to_commerc,xlamb2,ylamb2 060370002,34.13650,-117.9230,Indust, 9.250915, 7.503247,7.190049,7.660713,6.2324031, -22.4479346, -1.542679 060371002,34.17600,-118.3170,Comm_serv, 9.346770, 7.368975,9.3...
HW506Sol3.pdf
Path: Washington >> STAT >> 506 Spring, 2008
Description: Stat 506, Homework set #3 Due Monday April 21, 2008 From Casella and Berger. 3.3; 3.7; 3.9; 3.20 and 3.23 Solution to Casella and Berger 3.3 Let Xi be the indicator function of the event a car is passing during the i-th second, where we start counti...
hw5.pdf
Path: Washington >> STAT >> 506 Spring, 2008
Description: Computer Environments for Social Scientists CSSS 506 Professor: Mark S. Handcock Solutions to Homework 5 Due Tuesday, March 5, 2002 1) The data frame hills contains the results from the Scottish hill races. The data set is taken from Staudte and Shea...
512 Info.doc
Path: Washington >> STAT >> 512 Fall, 2008
Description: STAT512:STATISTICALINFERENCE AUTUMN2007 Instructor: MichaelPerlman,Dept.ofStatistics,Box354322 Office:B310PadelfordHall(mailboxinB313) Phone:5437735 email:michael@stat.washington.edu Officehours:afterclassorbyappointment MWF10:3011:20,Sieg225. Th10:3...
Rcode.txt
Path: Washington >> STAT >> 516 Fall, 2008
Description: Example R Sessionx <- 1:5xy <- 11:15x+yz <- c(10,6,3,5,1)zhelp(seq) seq(from=3,to=7,by=2)seq(3,17)seq(14,2)seq(1,2,.1)help(rep)rep(1,5)rep(1:5,3)temp <- c(2,1,5,9)temp.3 <- rep(temp,3)temp.3rep(c(seq(2,10,2),seq(1,9,2),2)temp.3[3] temp.3[5:7]temp.3[c...
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Path: Washington >> STAT >> 518 Spring, 2008
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Path: Washington >> STAT >> 518 Spring, 2008
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Path: Washington >> STAT >> 524 Spring, 2008
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CATHEEK STAT 534 HW #1.txt
Path: Washington >> STAT >> 534 Fall, 2008
Description: #-#-#Cathee Kneeling #STAT 534 #Homework #1, Due 4/11/2006 #--#-# Pull in source files directory <- \"http:/www.stat.washington.edu/catheek/stat534/HW#1\" lib.filename <- \"home1-make-data-lib-3-30-05.txt\" lib.pathname <- paste(directory, lib.filename, ...
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Path: Washington >> STAT >> 534 Fall, 2008
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Path: Washington >> STAT >> 535 Fall, 2008
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Path: Washington >> STAT >> 560 Fall, 2008
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Path: Washington >> STAT >> 560 Fall, 2008
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Path: Washington >> STAT >> 560 Fall, 2008
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Path: Washington >> STAT >> 567 Fall, 2008
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Path: Washington >> STAT >> 567 Fall, 2008
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Path: Washington >> STAT >> 567 Fall, 2008
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Path: Washington >> STAT >> 570 Fall, 2008
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Path: Washington >> STAT >> 576 Fall, 2008
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581.day1.08.pdf
Path: Washington >> STAT >> 581 Fall, 2008
Description: STATISTICS 581: Advanced Theory of Statistical Inference Fall, 2008 Time: Place: Professor: Oce: Phone: e-mail: Oce Hours: Texts: 10:30 - 11:20 MWF (lecture) MEB 245 Jon A. Wellner B320 Padelford 206-543-6207 jaw@stat.washington.edu 1:30 - 3:30 MWF...
ref.08.pdf
Path: Washington >> STAT >> 581 Fall, 2008
Description: References for Statistics 581, Fall 2008 Analysis: Bartle, R. G., The Elements of Integration. Rudin, W., Principles of Mathematical Analysis. Royden, H. L., Real Analysis. Luenberger, D., Optimization by Vector Space Methods. Probability: Will...
ch2.figs-epsf.pdf
Path: Washington >> STAT >> 581 Fall, 2008
Description: Statistics 581, Chapter 2 Empirical Distribution Function and Empirical Process Figures Wellner; 10/24/2008 1 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1 Figure 1: Uniform Empirical Distribution Function, n = 50. 1 0.75 0.5 0.25 0.2 -0.25 0.4 0.6...
exam1.06.pdf
Path: Washington >> STAT >> 581 Fall, 2008
Description: Statistics 581, Midterm Exam Wellner; 11/06/2006 This exam is to be taken without any books or notes. 1. (24 points) Dene any three of the following ve terms. (a) A uniformly integrable sequence of random variables. (b) Convergence in rth mean of a...
mt03.pdf
Path: Washington >> STAT >> 582 Fall, 2008
Description: Stat 582 W03 Midterm exam Please give as complete solutions as possible. More paper is available if needed. 1. Let (X,Y) be bivariate normal, mean zero, variance 1, correlation r. Find a minimal sufficient statistic for r. Is it complete? 2. Suppo...
final2.89.pdf
Path: Washington >> STAT >> 582 Fall, 2008
Description: STAT 582 FINAL EXAM 1. Let T (F) = m 2 = ( xdF(x)2 . Find, using the asymptotic theory for statistical functionals, the limiting F distribution of T (F n ) when m F 0. 2. Let (X 1 , Y 1 ), . . . , (X n , Y n ) be independent random variables with ...
gamma.pdf
Path: Washington >> STAT >> 582 Fall, 2008
Description: ...
hw1.sln.doc
Path: Washington >> STAT >> 583 Fall, 2008
Description: STAT583 Sp04 Homework1solution 1. (a)(c) E X i - m = 2 xdF(x) = d 2 2 s (1+ 2e) and (1- e)s + 3es ) = p p 0 E(X - m) 2 = s 2 (1+ 8e) .Standardasymptotictheoryhasthat n ( s2 - s 2 (1+ 8e) N(0,t 2 ) where 2 = E(X - m) 4 - E 2 (X - m) 2 = s 4 (3...
notes1.pdf
Path: Washington >> STAT >> 583 Fall, 2008
Description: STAT 583 SPRING 2008 Lecture Notes 1 Statistical Functionals The Gteux derivative of a statistical functional T(F) is the limit T (F + (G F ) T(F) . d1T(F;G F) = lim If Q( ) = T (F + (G F ) has a McLaurin expansion, we get an expansion (the von...
hw3.sln.pdf
Path: Washington >> STAT >> 583 Fall, 2008
Description: STAT 583 Sp08 Solutions, Homework 3 1. (a) T (F! ) = F + !(G \" F ) so IC(x) = x ! F . (b) By the same calculation as that for the median we get p ! 1(x < F !1 ( p) IC(x) = , x \" F !1 ( p) !1 f (F ( p) d 1 T (F + !(\" x # F) = (c) Using (b) d! 1 ...
HW2.pdf
Path: Washington >> STAT >> 583 Fall, 2008
Description: STAT 583 Sp08 Homework 2 Due April 16. 1. Let F be a cdf on [0,1], and define for ! > 1 T (F) = x\"[0,1] # ( F(x) ! F(x!) $ . (a) Compute the Gateux derivative of T at the uniform distribution U on [0,1]. (b) Show that nR1,n = n(T (Fn ) ! T (U ...
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Path: Washington >> STAT >> 592 Fall, 2008
Description: Homework problems (more problems will be added as we go along) Stat 592, W09 1. For a stationary random field Z(s); s !D \" R 2 , observed at sites s1,.,sn, derive the unbiased linear estimator with the smallest variance. Hint: Use a Lagrange multipli...
Lec1.ppt
Path: Washington >> STAT >> 592 Fall, 2008
Description: NRCSE SpatialStatisticalMethods peter@stat.washington.edu www.stat.washington.edu/peter/592 STAT592A(UW)526(UBCV) 8904(SFU) Coursecontent 1.Kriging 1.Gaussianregression 2.Simplekriging 3.Ordinaryanduniversalkriging 4.Effectofestimatedcovariance 5....
syllabus.pdf
Path: Washington >> STAT >> 593 Fall, 2008
Description: Course Syllabus: STAT 593 Modern Topics in Discrete Multivariate Analysis Spring Quarter 2008 Time and Location When: 10:30am-11:50am Tuesday and Thursday Where: TBA Instructor Adrian Dobra Oce: Padelford Hall B-303 E-mail: adobra@u.washington.edu ...
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Path: Washington >> HUM >> 202 Fall, 2008
Description: Lecture 14: October 27, 2006 Questions Reading Lyrical Ballads (1798), published Anonymously; 3 poems by Coleridge; 20 poems by Wordsworth Preface added in 1800 2nd edition Lyrical Ballads: not just an experiment but a poeti...
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Path: Washington >> HUM >> 204 Fall, 2008
Description: HUM 204 The Role of Perspective in History, Science, and Design Fall 2007 Lecture Meeting Times: Monday and Wednesday 12:30-1:50 Lecture Classroom: Kane 110 Instructors: Axel Roesler, Interaction Design Email: roesler@u.washington.edu Office Hours:...
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Path: Washington >> HUM >> 207 Winter, 2008
Description: Molecular and Cellular Endocrinology 161 (2000) 117 120 www.elsevier.com/locate/mce Spermatogonial transplantation an update for the millennium Lonnie D. Russella *, Michael D. Griswoldb a Department of Physiology, School of Medicine, Southern Il...
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Path: Washington >> HUM >> 208 Fall, 2008
Description: HUMANITIES 208: VIOLENCE, MYTH, AND MEMORY (SPRING 2008) Tuesday and Thursday 1:00-2:20, Johnson Hall 102 Francisco Benitez (Comp Lit) and Laurie Sears (History) TAs: Cheryll Alipio, Katrina Hagen, William Mitchell \"Violence, Myth, and Memory\" is bui...
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Path: Washington >> SWED >> 101 Fall, 2008
Description: Lecture 36 Anthropogenic Effects on Climate It is well-documented that globally averaged land and sea-surface temperatures have increased 0.5 C in the last century. Is this the beginning of manmade global warming? Two major anthropogenic forcings on ...
JFQA-402-JPPW-Appendices.pdf
Path: Washington >> T C >> 402 Winter, 2008
Description: Appendices to JFQA, Vol. 40, No. 2, June 2005, \"Horses and Rabbits? Trade-Off Theory and Optimal Capital Structure,\" by Nengjiu Ju, Robert Parrino, Allen M. Poteshman, and Michael S. Weisbach Horses and Rabbits? Trade-Off Theory and Optimal Capital ...
TC 403 Final.pdf
Path: Washington >> T C >> 403 Fall, 2008
Description: Kyle Kyros Starr TC 403 May 18, 2008 Do You Know the Way to San Jose? Exploring the Future During the course of this class, I have looked through a lens back at who I was in the past and evaluated that person against the person that I am now. In t...
MarxExcerpts.doc
Path: Washington >> T C >> 493 Fall, 2008
Description: Excerpts from Volume 1 of Capital (text from http:/www.marxists.org/archive/marx/works/1867-c1/index.htm) Introductory Note: Well discuss this in class, but just so you have it in front of you, Marx develops an analysis of capitalist production stres...
wave_properties.pdf
Path: Washington >> T C >> 505 Fall, 2008
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twodimensional.pdf
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intro.pdf
Path: Washington >> T C >> 509 Fall, 2008
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midterm_soln.pdf
Path: Washington >> T C >> 509 Fall, 2008
Description: GFD I: Winter 2007 Midterm Solutions Professor: Chris Bretherton TA: Tom Connolly \\) As A.\' ( ynOt/e g iawardCg- n\"a ir\",{ ina f r{ r,\\6rI V c\\.,r P *e*^p *ropi cc,t {cto nP- r {n 3ce t,( tt 17o r* lo t% scr rq- Av-ops {r\" ^ l o \\ o ,*b ...
hw7_soln.pdf
Path: Washington >> T C >> 509 Fall, 2008
Description: GFDI f HW io^t Sotto{ P.of, Ch.i e hn the\"r{o^ TA\' J[vu\"\'G^nolt1 f)\". No\',-\\incar SWt PV Conrrvr^4ion %:4) D+v _ .P/ )+ +- H *{-zv (,\"tlr, Ass,.^,- 5/r , , \\/3\'rth, q, 1-tr.) \'(\'i-r.)+H {t1)[r -efo,\"J TtlTm-ll1 x (+iir:Lr H T(Ti l-l 3=bll...
ch4.pdf
Path: Washington >> T C >> 516 Fall, 2008
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ps4.pdf
Path: Washington >> T C >> 521 Winter, 2008
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ps7.pdf
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Description: ...
sol.ps3.pdf
Path: Washington >> T C >> 521 Winter, 2008
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Path: Washington >> T C >> 598 Fall, 2008
Description: a) \\<\\:J \' .\\t \\ rf -: \'t .t\\, a : A O \\ \\ l R1 \\: q : F dl tr- (t) r-,\\i \'\\\'x A .i I 5 ;: H z f r\'l F (h F \'6; ar Y H ()Fr q aa frl Z F r\') 7 3 iJa -{ \'r1 a EK ,ii +) F v 6)x \\J e Y H ^ .H 14 F] (I a5 A 6 z ^ r- - z...
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Path: Washington >> T C >> 598 Fall, 2008
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Porter1976.pdf
Path: Washington >> THAI >> 302 Fall, 2008
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mt2_sol.pdf
Path: Washington >> IND E >> 101 Fall, 2008
Description: Name: _Section/TA Name: _ Atmospheric Science 101, Spring 2003 Midterm 2 Thursday 22 May 2003 Version A Multiple Choice (2 points each) Choose the best answer and mark it on a Scantron sheet. 1. Choose the words which best complete the following sent...
101.doc
Path: Washington >> IND E >> 101 Fall, 2008
Description: ABET Course Description IND E 101 Introduction to Industrial Engineering Elective Course Catalog course description: Examines the basic concepts and methods of industrial engineering through team-based hands-on activities. Explores the profession of ...
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Path: Washington >> IND E >> 250 Fall, 2008
Description: ABET Course Description IND E 250 Fundamentals of Engineering Economy Required Course Catalog course description: Basics of industrial cost analysis and accounting. Application of engineering economics to decision making. Analysis of engineering alte...
316.doc
Path: Washington >> IND E >> 316 Fall, 2008
Description: ABET Course Description IND E 316 Design of Experiments and Regression Analysis Required Course Catalog course description: Introduction to the analysis of data from planned experiments. Analysis of variance for multiple factors and applications of o...
351.doc
Path: Washington >> IND E >> 351 Spring, 2008
Description: ABET Course Description IND E 351 Human Factors in Design Elective Course Catalog Course Description: Engineering considerations of the abilities and limitations of the human aspect in the design of operational systems and components. Functional, psy...
410_UA_AAUH_syllabus.doc
Path: Washington >> IND E >> 410 Fall, 2008
Description: UNIVERSITY OF OREGON Department of History Spring, 2006 Professor Quintard Taylor Office: 364 McKenzie Hall Email: qtaylor@uoregon.edu Phone: 346-6160 Web: http:/faculty.washington.edu/qtaylor/ Office Hours: 2:00-3:30 Tuesday History 410/510 Urban Ar...
