Math 426: Probability MWF 1pm, Gasson 310 Homework 4 Selected Solutions

Size: px
Start display at page:

Download "Math 426: Probability MWF 1pm, Gasson 310 Homework 4 Selected Solutions"

Transcription

1 Exercses from Ross, 3, : Math 26: Probablty MWF pm, Gasson 30 Homework Selected Solutons 3, p. 05 Problems 76, 86 3, p. 06 Theoretcal exercses 3, 6, p. 63 Problems 5, 0, 20, p. 69 Theoretcal exercses 2, 3, 3, Problem 86. Let S {,..., n}m and suppose that A and B are, ndependently, equally lkely to be any of the 2 n subsets ncludng and S of S. a Show that P A B 3/ n. b Show that P A B 3/ n. Soluton a. Followng the hnt n the book, we condton on the number of elements n B: let NB ndcate the number of elements n B. Evdently, NB {0,..., n}. Because the whole sample space s the dsjont unon of the events n whch B has k elements as k goes from to n, by Bayes Theorem we have P A B P A B NB P NB. Moreover, gven that NB, there are 2 subsets of B, and 2 n subsets n total. Thus P A B NB 2 2 n. Fnally, there are n subsets of S of sze, so we have Puttng these together, we have P NB n 2 n. P A B P A B NB P NB n 2 n 2 n n n 2 2 n n 2 n + n 2n n 3. Soluton b. The condton A B s equvalent to A B c. The probablty P A B c can be computed wth the same method as above: Bayes Theorem agan apples condtonng on NB c, the

2 number of elements n B c, and P NB c P NB n. Thus we have P A B c P A B c NB c P NB c n 2 n 2 n n j0 n j n 2 n n n. 2 n n j 2 n j + n 2n 3 n 2 n n 3, Theoretcal exercse 3. A communty conssts of m famles, where n of them have chldren, for each,..., k, so that n m. Consder the followng two methods for choosng a random chld: Choose one of the m famles at random, then randomly choose a chld from that famly. 2 Choose one of the n chldren at random. Show that method s more lkely than method 2 to result n the choce of a frstborn chld. Soluton. Let F ndcate the event that a frstborn chld s chosen. We compute P F for each method, startng wth method. Let E ndcate the event that we start by choosng a famly wth chldren. Snce we are choosng one of m famles randomly and n of them have chldren, we have P E n /m. Snce E,..., E k are dsjont sets whose unon s the whole sample space, by Bayes Theorem we have P F P F E P E. Moreover, gven that we ve chosen a famly wth chldren, the probablty that a randomly chosen chld s frstborn s /. Thus P F E /. Puttng ths altogether, we have P F P F E P E n m m Call ths probablty f. Now we compute P F for method 2: because there are n chldren and m frstborn, we have P F m. n Call ths probablty f 2. We clam that f f 2. Ths nequalty s equvalent to n mf n mf 2, and snce m n the latter s equvalent to n 2 n n. Introducng a second ndex for convenence, ths s equvalent to n jn j n n j. j j 2 n.

3 It remans to demonstrate ths nequalty. We wll wrte LHS and RHS for the lefthand and rghthand sdes, respectvely, of the last nequalty above. By expandng these products, we have RHS n n j n n j n n j + 2n n j j j j <j n 2 + 2n n j <j n j LHS jn j n n j j n n j + j + j n n j j j j <j n 2 + j + j n n j. <j Evdently, we wll have LHS RHS provded we show that + j 2, for all postve ntegers, j. j Because j s an nteger, we have j 2 0, whch when expanded and rearranged gves 2 +j 2 2j. Snce and j are postve, ths mples that 2 + j 2 /j 2, whch s precsely + j 2 as desred. j, Problem 20. A gamblng book recommends the followng wnnng strategy for the game of roulette: Bet $ on red. If red appears whch has probablty 8/38, then take the $ proft and qut. If red does not appear and you lose ths bet whch probablty 20/38 of occurng, make addtonal $ bets on red on each of the next two spns of the roulette wheel and then qut. Let X denote your wnnngs when you qut. a Fnd P X > 0. b Are you convnced that the strategy s ndeed wnnng? Explan. c Fnd E[X]. Soluton a. Let R ndcate the event that red appears on the frst spn of the wheel. If red appears on the frst spn, than our strategy produces a proft of $, so that P X > 0 R. Let R ndcate the event that there are reds on the second and thrd spns of the roulette wheel, for 0,, 2. Snce R, R c R 0, R c R, R c R 2 are dsjont sets whose unon s the whole sample space, by Bayes Theorem we have Our strategy dctates: P X > 0 P X > 0 RP R + P X > 0 R c R 0 P R c R 0 + P X > 0 R c R P R c R + P X > 0 R c R 2 P R c R 2 f ω R 3 f ω R c R 0 Xω f ω R c R f ω R c R 2 Thus P X > 0 R P X > 0 R c R 2 and P X > 0 R c R 0 P X > 0 R c R 0, and we conclude that P X > 0 P R + P R c R

4 Soluton b. The answer here depends a bt on what s meant by wnnng. The gamblng book s correct n the sense that P X > 0 > /2, so that we are more lkely than not to make money. However, f we play ths game over and over, repeatedly, whether we make money n the long run whch many readers would farly nterpret wnnng to mean depends on how much we we wn or lose as well. That s, a reasonable nterpretaton of wnnng depends on E[X], and not on P X > 0. Snce we compute ths quantty below, and E[X] < 0, one could farly nterpret ths strategy to be losng. Soluton c. The possble values for X are recorded above, and they are, 3, and. Thus we have E[X] P X 3 P X 3 P X P R R c R 2 3P R c R 0 P R c R If we perform ths strategy 000 tmes, we should expect fnal wnnngs to be approxmately ,.e. we expect to lose $08., Theoretcal exercse. The random varable X s sad to have the Yule-Smons dstrbuton f P X n, for n. nn + a Show that the precedng s actually a probablty mass functon,.e. P X n. b Show that E[X] 2. c Show that E[X 2 ]. Soluton a. We frst smplfy the expresson for P X n: nn + nn + n n n + 2 Now the partal sums can be computed drectly: P X n nn + n 2 n + n N N n N 2 n + 2 N + 2 N N + 2. N n n. + 2 N + N + 2

5 To fnsh, the nfnte sum s gven by the lmt of the partal sums: P X n lm N P X n lm N 2 N N + 2. Soluton b. We compute the expected value drectly: E[X] n P X n n + lm N lm N N lm N n + 2 N + 2 n nn + n + lm N 2 2 N Soluton c. For the second moment, we have: E[X 2 ] n 2 P X n n n +. n 2 nn + Snce n + 2n and 3n for all n, we have n + 6n 2. Because the seres n 6n 2 dverges by the p-test, or because the harmonc seres dverges, the comparson 2 3 n test now mples that n n + dverges. Thus E[X2 ]. The followng problem appears as Theoretcal exercse n a prevous edton of the textbook:, Theoretcal exercse. For a nonnegatve nteger-valued random varable N, show that E[N] P N. Soluton. Because the event N s the dsjont unon of the events N j, for j, +,..., we have P N P N j and j P N P N j. j 5

6 For a fxed value of j, P N j appears n the double sum once for each value of from to j. That s, nterchangng the order of summaton gves j P N P N j j P N j. j j Because N s non-negatve and nteger-valued, ths last sum s the same as x P N x,.e. E[N]. x:px>0 6

Expected Value and Variance

Expected Value and Variance MATH 38 Expected Value and Varance Dr. Neal, WKU We now shall dscuss how to fnd the average and standard devaton of a random varable X. Expected Value Defnton. The expected value (or average value, or

More information

REAL ANALYSIS I HOMEWORK 1

REAL ANALYSIS I HOMEWORK 1 REAL ANALYSIS I HOMEWORK CİHAN BAHRAN The questons are from Tao s text. Exercse 0.0.. If (x α ) α A s a collecton of numbers x α [0, + ] such that x α

More information

First day August 1, Problems and Solutions

First day August 1, Problems and Solutions FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 August 4, 997, Plovdv, BULGARIA Frst day August, 997 Problems and Solutons Problem. Let {ε n } n= be a sequence of postve

More information

SELECTED PROOFS. DeMorgan s formulas: The first one is clear from Venn diagram, or the following truth table:

SELECTED PROOFS. DeMorgan s formulas: The first one is clear from Venn diagram, or the following truth table: SELECTED PROOFS DeMorgan s formulas: The frst one s clear from Venn dagram, or the followng truth table: A B A B A B Ā B Ā B T T T F F F F T F T F F T F F T T F T F F F F F T T T T The second one can be

More information

Applied Stochastic Processes

Applied Stochastic Processes STAT455/855 Fall 23 Appled Stochastc Processes Fnal Exam, Bref Solutons 1. (15 marks) (a) (7 marks) The dstrbuton of Y s gven by ( ) ( ) y 2 1 5 P (Y y) for y 2, 3,... The above follows because each of

More information

princeton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg

princeton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg prnceton unv. F 17 cos 521: Advanced Algorthm Desgn Lecture 7: LP Dualty Lecturer: Matt Wenberg Scrbe: LP Dualty s an extremely useful tool for analyzng structural propertes of lnear programs. Whle there

More information

Foundations of Arithmetic

Foundations of Arithmetic Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an

More information

and problem sheet 2

and problem sheet 2 -8 and 5-5 problem sheet Solutons to the followng seven exercses and optonal bonus problem are to be submtted through gradescope by :0PM on Wednesday th September 08. There are also some practce problems,

More information

Maximizing the number of nonnegative subsets

Maximizing the number of nonnegative subsets Maxmzng the number of nonnegatve subsets Noga Alon Hao Huang December 1, 213 Abstract Gven a set of n real numbers, f the sum of elements of every subset of sze larger than k s negatve, what s the maxmum

More information

NUMERICAL DIFFERENTIATION

NUMERICAL DIFFERENTIATION NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the

More information

Exercises of Chapter 2

Exercises of Chapter 2 Exercses of Chapter Chuang-Cheh Ln Department of Computer Scence and Informaton Engneerng, Natonal Chung Cheng Unversty, Mng-Hsung, Chay 61, Tawan. Exercse.6. Suppose that we ndependently roll two standard

More information

HMMT February 2016 February 20, 2016

HMMT February 2016 February 20, 2016 HMMT February 016 February 0, 016 Combnatorcs 1. For postve ntegers n, let S n be the set of ntegers x such that n dstnct lnes, no three concurrent, can dvde a plane nto x regons (for example, S = {3,

More information

MATH 281A: Homework #6

MATH 281A: Homework #6 MATH 28A: Homework #6 Jongha Ryu Due date: November 8, 206 Problem. (Problem 2..2. Soluton. If X,..., X n Bern(p, then T = X s a complete suffcent statstc. Our target s g(p = p, and the nave guess suggested

More information

ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)

ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U) Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of

More information

More metrics on cartesian products

More metrics on cartesian products More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of

More information

COS 521: Advanced Algorithms Game Theory and Linear Programming

COS 521: Advanced Algorithms Game Theory and Linear Programming COS 521: Advanced Algorthms Game Theory and Lnear Programmng Moses Charkar February 27, 2013 In these notes, we ntroduce some basc concepts n game theory and lnear programmng (LP). We show a connecton

More information

Problem Set 9 Solutions

Problem Set 9 Solutions Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem

More information

Math 217 Fall 2013 Homework 2 Solutions

Math 217 Fall 2013 Homework 2 Solutions Math 17 Fall 013 Homework Solutons Due Thursday Sept. 6, 013 5pm Ths homework conssts of 6 problems of 5 ponts each. The total s 30. You need to fully justfy your answer prove that your functon ndeed has

More information

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could

More information

Prof. Dr. I. Nasser Phys 630, T Aug-15 One_dimensional_Ising_Model

Prof. Dr. I. Nasser Phys 630, T Aug-15 One_dimensional_Ising_Model EXACT OE-DIMESIOAL ISIG MODEL The one-dmensonal Isng model conssts of a chan of spns, each spn nteractng only wth ts two nearest neghbors. The smple Isng problem n one dmenson can be solved drectly n several

More information

MAE140 - Linear Circuits - Fall 13 Midterm, October 31

MAE140 - Linear Circuits - Fall 13 Midterm, October 31 Instructons ME140 - Lnear Crcuts - Fall 13 Mdterm, October 31 () Ths exam s open book. You may use whatever wrtten materals you choose, ncludng your class notes and textbook. You may use a hand calculator

More information

2.3 Nilpotent endomorphisms

2.3 Nilpotent endomorphisms s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms

More information

A note on almost sure behavior of randomly weighted sums of φ-mixing random variables with φ-mixing weights

A note on almost sure behavior of randomly weighted sums of φ-mixing random variables with φ-mixing weights ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 7, Number 2, December 203 Avalable onlne at http://acutm.math.ut.ee A note on almost sure behavor of randomly weghted sums of φ-mxng

More information

Exercise Solutions to Real Analysis

Exercise Solutions to Real Analysis xercse Solutons to Real Analyss Note: References refer to H. L. Royden, Real Analyss xersze 1. Gven any set A any ɛ > 0, there s an open set O such that A O m O m A + ɛ. Soluton 1. If m A =, then there

More information

Randomness and Computation

Randomness and Computation Randomness and Computaton or, Randomzed Algorthms Mary Cryan School of Informatcs Unversty of Ednburgh RC 208/9) Lecture 0 slde Balls n Bns m balls, n bns, and balls thrown unformly at random nto bns usually

More information

11 Tail Inequalities Markov s Inequality. Lecture 11: Tail Inequalities [Fa 13]

11 Tail Inequalities Markov s Inequality. Lecture 11: Tail Inequalities [Fa 13] Algorthms Lecture 11: Tal Inequaltes [Fa 13] If you hold a cat by the tal you learn thngs you cannot learn any other way. Mark Twan 11 Tal Inequaltes The smple recursve structure of skp lsts made t relatvely

More information

MATH 5707 HOMEWORK 4 SOLUTIONS 2. 2 i 2p i E(X i ) + E(Xi 2 ) ä i=1. i=1

MATH 5707 HOMEWORK 4 SOLUTIONS 2. 2 i 2p i E(X i ) + E(Xi 2 ) ä i=1. i=1 MATH 5707 HOMEWORK 4 SOLUTIONS CİHAN BAHRAN 1. Let v 1,..., v n R m, all lengths v are not larger than 1. Let p 1,..., p n [0, 1] be arbtrary and set w = p 1 v 1 + + p n v n. Then there exst ε 1,..., ε

More information

Lecture 4: Universal Hash Functions/Streaming Cont d

Lecture 4: Universal Hash Functions/Streaming Cont d CSE 5: Desgn and Analyss of Algorthms I Sprng 06 Lecture 4: Unversal Hash Functons/Streamng Cont d Lecturer: Shayan Oves Gharan Aprl 6th Scrbe: Jacob Schreber Dsclamer: These notes have not been subjected

More information

Errors for Linear Systems

Errors for Linear Systems Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch

More information

Appendix B. Criterion of Riemann-Stieltjes Integrability

Appendix B. Criterion of Riemann-Stieltjes Integrability Appendx B. Crteron of Remann-Steltes Integrablty Ths note s complementary to [R, Ch. 6] and [T, Sec. 3.5]. The man result of ths note s Theorem B.3, whch provdes the necessary and suffcent condtons for

More information

Lecture Notes on Linear Regression

Lecture Notes on Linear Regression Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume

More information

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number

More information

Homework Notes Week 7

Homework Notes Week 7 Homework Notes Week 7 Math 4 Sprng 4 #4 (a Complete the proof n example 5 that s an nner product (the Frobenus nner product on M n n (F In the example propertes (a and (d have already been verfed so we

More information

FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP

FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class

More information

Econ107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)

Econ107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4) I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes

More information

Discussion 11 Summary 11/20/2018

Discussion 11 Summary 11/20/2018 Dscusson 11 Summary 11/20/2018 1 Quz 8 1. Prove for any sets A, B that A = A B ff B A. Soluton: There are two drectons we need to prove: (a) A = A B B A, (b) B A A = A B. (a) Frst, we prove A = A B B A.

More information

Some basic inequalities. Definition. Let V be a vector space over the complex numbers. An inner product is given by a function, V V C

Some basic inequalities. Definition. Let V be a vector space over the complex numbers. An inner product is given by a function, V V C Some basc nequaltes Defnton. Let V be a vector space over the complex numbers. An nner product s gven by a functon, V V C (x, y) x, y satsfyng the followng propertes (for all x V, y V and c C) (1) x +

More information

1 Definition of Rademacher Complexity

1 Definition of Rademacher Complexity COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture #9 Scrbe: Josh Chen March 5, 2013 We ve spent the past few classes provng bounds on the generalzaton error of PAClearnng algorths for the

More information

FINITELY-GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN

FINITELY-GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN FINITELY-GENERTED MODULES OVER PRINCIPL IDEL DOMIN EMMNUEL KOWLSKI Throughout ths note, s a prncpal deal doman. We recall the classfcaton theorem: Theorem 1. Let M be a fntely-generated -module. (1) There

More information

Bernoulli Numbers and Polynomials

Bernoulli Numbers and Polynomials Bernoull Numbers and Polynomals T. Muthukumar tmk@tk.ac.n 17 Jun 2014 The sum of frst n natural numbers 1, 2, 3,..., n s n n(n + 1 S 1 (n := m = = n2 2 2 + n 2. Ths formula can be derved by notng that

More information

Stanford University CS254: Computational Complexity Notes 7 Luca Trevisan January 29, Notes for Lecture 7

Stanford University CS254: Computational Complexity Notes 7 Luca Trevisan January 29, Notes for Lecture 7 Stanford Unversty CS54: Computatonal Complexty Notes 7 Luca Trevsan January 9, 014 Notes for Lecture 7 1 Approxmate Countng wt an N oracle We complete te proof of te followng result: Teorem 1 For every

More information

Math 261 Exercise sheet 2

Math 261 Exercise sheet 2 Math 261 Exercse sheet 2 http://staff.aub.edu.lb/~nm116/teachng/2017/math261/ndex.html Verson: September 25, 2017 Answers are due for Monday 25 September, 11AM. The use of calculators s allowed. Exercse

More information

Affine transformations and convexity

Affine transformations and convexity Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/

More information

Lecture 10 Support Vector Machines II

Lecture 10 Support Vector Machines II Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed

More information

8.6 The Complex Number System

8.6 The Complex Number System 8.6 The Complex Number System Earler n the chapter, we mentoned that we cannot have a negatve under a square root, snce the square of any postve or negatve number s always postve. In ths secton we want

More information

Lecture 3: Probability Distributions

Lecture 3: Probability Distributions Lecture 3: Probablty Dstrbutons Random Varables Let us begn by defnng a sample space as a set of outcomes from an experment. We denote ths by S. A random varable s a functon whch maps outcomes nto the

More information

Lecture Space-Bounded Derandomization

Lecture Space-Bounded Derandomization Notes on Complexty Theory Last updated: October, 2008 Jonathan Katz Lecture Space-Bounded Derandomzaton 1 Space-Bounded Derandomzaton We now dscuss derandomzaton of space-bounded algorthms. Here non-trval

More information

Problem Do any of the following determine homomorphisms from GL n (C) to GL n (C)?

Problem Do any of the following determine homomorphisms from GL n (C) to GL n (C)? Homework 8 solutons. Problem 16.1. Whch of the followng defne homomomorphsms from C\{0} to C\{0}? Answer. a) f 1 : z z Yes, f 1 s a homomorphsm. We have that z s the complex conjugate of z. If z 1,z 2

More information

Math 702 Midterm Exam Solutions

Math 702 Midterm Exam Solutions Math 702 Mdterm xam Solutons The terms measurable, measure, ntegrable, and almost everywhere (a.e.) n a ucldean space always refer to Lebesgue measure m. Problem. [6 pts] In each case, prove the statement

More information

Module 2. Random Processes. Version 2 ECE IIT, Kharagpur

Module 2. Random Processes. Version 2 ECE IIT, Kharagpur Module Random Processes Lesson 6 Functons of Random Varables After readng ths lesson, ou wll learn about cdf of functon of a random varable. Formula for determnng the pdf of a random varable. Let, X be

More information

Stanford University CS359G: Graph Partitioning and Expanders Handout 4 Luca Trevisan January 13, 2011

Stanford University CS359G: Graph Partitioning and Expanders Handout 4 Luca Trevisan January 13, 2011 Stanford Unversty CS359G: Graph Parttonng and Expanders Handout 4 Luca Trevsan January 3, 0 Lecture 4 In whch we prove the dffcult drecton of Cheeger s nequalty. As n the past lectures, consder an undrected

More information

PROBABILITY PRIMER. Exercise Solutions

PROBABILITY PRIMER. Exercise Solutions PROBABILITY PRIMER Exercse Solutons 1 Probablty Prmer, Exercse Solutons, Prncples of Econometrcs, e EXERCISE P.1 (b) X s a random varable because attendance s not known pror to the outdoor concert. Before

More information

Linear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space.

Linear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space. Lnear, affne, and convex sets and hulls In the sequel, unless otherwse specfed, X wll denote a real vector space. Lnes and segments. Gven two ponts x, y X, we defne xy = {x + t(y x) : t R} = {(1 t)x +

More information

Homework Assignment 3 Due in class, Thursday October 15

Homework Assignment 3 Due in class, Thursday October 15 Homework Assgnment 3 Due n class, Thursday October 15 SDS 383C Statstcal Modelng I 1 Rdge regresson and Lasso 1. Get the Prostrate cancer data from http://statweb.stanford.edu/~tbs/elemstatlearn/ datasets/prostate.data.

More information

Finding Dense Subgraphs in G(n, 1/2)

Finding Dense Subgraphs in G(n, 1/2) Fndng Dense Subgraphs n Gn, 1/ Atsh Das Sarma 1, Amt Deshpande, and Rav Kannan 1 Georga Insttute of Technology,atsh@cc.gatech.edu Mcrosoft Research-Bangalore,amtdesh,annan@mcrosoft.com Abstract. Fndng

More information

Self-complementing permutations of k-uniform hypergraphs

Self-complementing permutations of k-uniform hypergraphs Dscrete Mathematcs Theoretcal Computer Scence DMTCS vol. 11:1, 2009, 117 124 Self-complementng permutatons of k-unform hypergraphs Artur Szymańsk A. Paweł Wojda Faculty of Appled Mathematcs, AGH Unversty

More information

Homework 9 Solutions. 1. (Exercises from the book, 6 th edition, 6.6, 1-3.) Determine the number of distinct orderings of the letters given:

Homework 9 Solutions. 1. (Exercises from the book, 6 th edition, 6.6, 1-3.) Determine the number of distinct orderings of the letters given: Homework 9 Solutons PROBLEM ONE 1 (Exercses from the book, th edton,, 1-) Determne the number of dstnct orderngs of the letters gven: (a) GUIDE Soluton: 5! (b) SCHOOL Soluton:! (c) SALESPERSONS Soluton:

More information

Physics 4B. A positive value is obtained, so the current is counterclockwise around the circuit.

Physics 4B. A positive value is obtained, so the current is counterclockwise around the circuit. Physcs 4B Solutons to Chapter 7 HW Chapter 7: Questons:, 8, 0 Problems:,,, 45, 48,,, 7, 9 Queston 7- (a) no (b) yes (c) all te Queston 7-8 0 μc Queston 7-0, c;, a;, d; 4, b Problem 7- (a) Let be the current

More information

MMA and GCMMA two methods for nonlinear optimization

MMA and GCMMA two methods for nonlinear optimization MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons

More information

One Dimension Again. Chapter Fourteen

One Dimension Again. Chapter Fourteen hapter Fourteen One Dmenson Agan 4 Scalar Lne Integrals Now we agan consder the dea of the ntegral n one dmenson When we were ntroduced to the ntegral back n elementary school, we consdered only functons

More information

E Tail Inequalities. E.1 Markov s Inequality. Non-Lecture E: Tail Inequalities

E Tail Inequalities. E.1 Markov s Inequality. Non-Lecture E: Tail Inequalities Algorthms Non-Lecture E: Tal Inequaltes If you hold a cat by the tal you learn thngs you cannot learn any other way. Mar Twan E Tal Inequaltes The smple recursve structure of sp lsts made t relatvely easy

More information

1. Estimation, Approximation and Errors Percentages Polynomials and Formulas Identities and Factorization 52

1. Estimation, Approximation and Errors Percentages Polynomials and Formulas Identities and Factorization 52 ontents ommonly Used Formulas. Estmaton, pproxmaton and Errors. Percentages. Polynomals and Formulas 8. Identtes and Factorzaton. Equatons and Inequaltes 66 6. Rate and Rato 8 7. Laws of Integral Indces

More information

Complete subgraphs in multipartite graphs

Complete subgraphs in multipartite graphs Complete subgraphs n multpartte graphs FLORIAN PFENDER Unverstät Rostock, Insttut für Mathematk D-18057 Rostock, Germany Floran.Pfender@un-rostock.de Abstract Turán s Theorem states that every graph G

More information

BOUNDEDNESS OF THE RIESZ TRANSFORM WITH MATRIX A 2 WEIGHTS

BOUNDEDNESS OF THE RIESZ TRANSFORM WITH MATRIX A 2 WEIGHTS BOUNDEDNESS OF THE IESZ TANSFOM WITH MATIX A WEIGHTS Introducton Let L = L ( n, be the functon space wth norm (ˆ f L = f(x C dx d < For a d d matrx valued functon W : wth W (x postve sem-defnte for all

More information

Communication Complexity 16:198: February Lecture 4. x ij y ij

Communication Complexity 16:198: February Lecture 4. x ij y ij Communcaton Complexty 16:198:671 09 February 2010 Lecture 4 Lecturer: Troy Lee Scrbe: Rajat Mttal 1 Homework problem : Trbes We wll solve the thrd queston n the homework. The goal s to show that the nondetermnstc

More information

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0 MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector

More information

Using T.O.M to Estimate Parameter of distributions that have not Single Exponential Family

Using T.O.M to Estimate Parameter of distributions that have not Single Exponential Family IOSR Journal of Mathematcs IOSR-JM) ISSN: 2278-5728. Volume 3, Issue 3 Sep-Oct. 202), PP 44-48 www.osrjournals.org Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly Jubran

More information

APPROXIMATE PRICES OF BASKET AND ASIAN OPTIONS DUPONT OLIVIER. Premia 14

APPROXIMATE PRICES OF BASKET AND ASIAN OPTIONS DUPONT OLIVIER. Premia 14 APPROXIMAE PRICES OF BASKE AND ASIAN OPIONS DUPON OLIVIER Prema 14 Contents Introducton 1 1. Framewor 1 1.1. Baset optons 1.. Asan optons. Computng the prce 3. Lower bound 3.1. Closed formula for the prce

More information

A combinatorial problem associated with nonograms

A combinatorial problem associated with nonograms A combnatoral problem assocated wth nonograms Jessca Benton Ron Snow Nolan Wallach March 21, 2005 1 Introducton. Ths work was motvated by a queston posed by the second named author to the frst named author

More information

Edge Isoperimetric Inequalities

Edge Isoperimetric Inequalities November 7, 2005 Ross M. Rchardson Edge Isopermetrc Inequaltes 1 Four Questons Recall that n the last lecture we looked at the problem of sopermetrc nequaltes n the hypercube, Q n. Our noton of boundary

More information

Distribution of subgraphs of random regular graphs

Distribution of subgraphs of random regular graphs Dstrbuton of subgraphs of random regular graphs Zhcheng Gao Faculty of Busness Admnstraton Unversty of Macau Macau Chna zcgao@umac.mo N. C. Wormald Department of Combnatorcs and Optmzaton Unversty of Waterloo

More information

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com PhscsAndMathsTutor.com phscsandmathstutor.com June 005 5. The random varable X has probablt functon k, = 1,, 3, P( X = ) = k ( + 1), = 4, 5, where k s a constant. (a) Fnd the value of k. (b) Fnd the eact

More information

Appendix for Causal Interaction in Factorial Experiments: Application to Conjoint Analysis

Appendix for Causal Interaction in Factorial Experiments: Application to Conjoint Analysis A Appendx for Causal Interacton n Factoral Experments: Applcaton to Conjont Analyss Mathematcal Appendx: Proofs of Theorems A. Lemmas Below, we descrbe all the lemmas, whch are used to prove the man theorems

More information

Google PageRank with Stochastic Matrix

Google PageRank with Stochastic Matrix Google PageRank wth Stochastc Matrx Md. Sharq, Puranjt Sanyal, Samk Mtra (M.Sc. Applcatons of Mathematcs) Dscrete Tme Markov Chan Let S be a countable set (usually S s a subset of Z or Z d or R or R d

More information

Expectation propagation

Expectation propagation Expectaton propagaton Lloyd Ellott May 17, 2011 Suppose p(x) s a pdf and we have a factorzaton p(x) = 1 Z n f (x). (1) =1 Expectaton propagaton s an nference algorthm desgned to approxmate the factors

More information

The Expectation-Maximization Algorithm

The Expectation-Maximization Algorithm The Expectaton-Maxmaton Algorthm Charles Elan elan@cs.ucsd.edu November 16, 2007 Ths chapter explans the EM algorthm at multple levels of generalty. Secton 1 gves the standard hgh-level verson of the algorthm.

More information

Solutions to the 71st William Lowell Putnam Mathematical Competition Saturday, December 4, 2010

Solutions to the 71st William Lowell Putnam Mathematical Competition Saturday, December 4, 2010 Solutons to the 7st Wllam Lowell Putnam Mathematcal Competton Saturday, December 4, 2 Kran Kedlaya and Lenny Ng A The largest such k s n+ 2 n 2. For n even, ths value s acheved by the partton {,n},{2,n

More information

An (almost) unbiased estimator for the S-Gini index

An (almost) unbiased estimator for the S-Gini index An (almost unbased estmator for the S-Gn ndex Thomas Demuynck February 25, 2009 Abstract Ths note provdes an unbased estmator for the absolute S-Gn and an almost unbased estmator for the relatve S-Gn for

More information

Open Systems: Chemical Potential and Partial Molar Quantities Chemical Potential

Open Systems: Chemical Potential and Partial Molar Quantities Chemical Potential Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,

More information

Section 8.3 Polar Form of Complex Numbers

Section 8.3 Polar Form of Complex Numbers 80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the

More information

Difference Equations

Difference Equations Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1

More information

Lecture 3. Ax x i a i. i i

Lecture 3. Ax x i a i. i i 18.409 The Behavor of Algorthms n Practce 2/14/2 Lecturer: Dan Spelman Lecture 3 Scrbe: Arvnd Sankar 1 Largest sngular value In order to bound the condton number, we need an upper bound on the largest

More information

CALCULUS CLASSROOM CAPSULES

CALCULUS CLASSROOM CAPSULES CALCULUS CLASSROOM CAPSULES SESSION S86 Dr. Sham Alfred Rartan Valley Communty College salfred@rartanval.edu 38th AMATYC Annual Conference Jacksonvlle, Florda November 8-, 202 2 Calculus Classroom Capsules

More information

Complex Numbers Alpha, Round 1 Test #123

Complex Numbers Alpha, Round 1 Test #123 Complex Numbers Alpha, Round Test #3. Wrte your 6-dgt ID# n the I.D. NUMBER grd, left-justfed, and bubble. Check that each column has only one number darkened.. In the EXAM NO. grd, wrte the 3-dgt Test

More information

STAT 511 FINAL EXAM NAME Spring 2001

STAT 511 FINAL EXAM NAME Spring 2001 STAT 5 FINAL EXAM NAME Sprng Instructons: Ths s a closed book exam. No notes or books are allowed. ou may use a calculator but you are not allowed to store notes or formulas n the calculator. Please wrte

More information

Lecture 21: Numerical methods for pricing American type derivatives

Lecture 21: Numerical methods for pricing American type derivatives Lecture 21: Numercal methods for prcng Amercan type dervatves Xaoguang Wang STAT 598W Aprl 10th, 2014 (STAT 598W) Lecture 21 1 / 26 Outlne 1 Fnte Dfference Method Explct Method Penalty Method (STAT 598W)

More information

find (x): given element x, return the canonical element of the set containing x;

find (x): given element x, return the canonical element of the set containing x; COS 43 Sprng, 009 Dsjont Set Unon Problem: Mantan a collecton of dsjont sets. Two operatons: fnd the set contanng a gven element; unte two sets nto one (destructvely). Approach: Canoncal element method:

More information

Hila Etzion. Min-Seok Pang

Hila Etzion. Min-Seok Pang RESERCH RTICLE COPLEENTRY ONLINE SERVICES IN COPETITIVE RKETS: INTINING PROFITILITY IN THE PRESENCE OF NETWORK EFFECTS Hla Etzon Department of Technology and Operatons, Stephen. Ross School of usness,

More information

A new construction of 3-separable matrices via an improved decoding of Macula s construction

A new construction of 3-separable matrices via an improved decoding of Macula s construction Dscrete Optmzaton 5 008 700 704 Contents lsts avalable at ScenceDrect Dscrete Optmzaton journal homepage: wwwelsevercom/locate/dsopt A new constructon of 3-separable matrces va an mproved decodng of Macula

More information

10. Canonical Transformations Michael Fowler

10. Canonical Transformations Michael Fowler 10. Canoncal Transformatons Mchael Fowler Pont Transformatons It s clear that Lagrange s equatons are correct for any reasonable choce of parameters labelng the system confguraton. Let s call our frst

More information

Limited Dependent Variables

Limited Dependent Variables Lmted Dependent Varables. What f the left-hand sde varable s not a contnuous thng spread from mnus nfnty to plus nfnty? That s, gven a model = f (, β, ε, where a. s bounded below at zero, such as wages

More information

Engineering Risk Benefit Analysis

Engineering Risk Benefit Analysis Engneerng Rsk Beneft Analyss.55, 2.943, 3.577, 6.938, 0.86, 3.62, 6.862, 22.82, ESD.72, ESD.72 RPRA 2. Elements of Probablty Theory George E. Apostolaks Massachusetts Insttute of Technology Sprng 2007

More information

Structure and Drive Paul A. Jensen Copyright July 20, 2003

Structure and Drive Paul A. Jensen Copyright July 20, 2003 Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.

More information

Canonical transformations

Canonical transformations Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,

More information

MAE140 - Linear Circuits - Winter 16 Final, March 16, 2016

MAE140 - Linear Circuits - Winter 16 Final, March 16, 2016 ME140 - Lnear rcuts - Wnter 16 Fnal, March 16, 2016 Instructons () The exam s open book. You may use your class notes and textbook. You may use a hand calculator wth no communcaton capabltes. () You have

More information

), it produces a response (output function g (x)

), it produces a response (output function g (x) Lnear Systems Revew Notes adapted from notes by Mchael Braun Typcally n electrcal engneerng, one s concerned wth functons of tme, such as a voltage waveform System descrpton s therefore defned n the domans

More information

THE SUMMATION NOTATION Ʃ

THE SUMMATION NOTATION Ʃ Sngle Subscrpt otaton THE SUMMATIO OTATIO Ʃ Most of the calculatons we perform n statstcs are repettve operatons on lsts of numbers. For example, we compute the sum of a set of numbers, or the sum of the

More information

20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The first idea is connectedness.

20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The first idea is connectedness. 20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The frst dea s connectedness. Essentally, we want to say that a space cannot be decomposed

More information

The Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction

The Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also

More information

International Mathematical Olympiad. Preliminary Selection Contest 2012 Hong Kong. Outline of Solutions

International Mathematical Olympiad. Preliminary Selection Contest 2012 Hong Kong. Outline of Solutions Internatonal Mathematcal Olympad Prelmnary Selecton ontest Hong Kong Outlne of Solutons nswers: 7 4 7 4 6 5 9 6 99 7 6 6 9 5544 49 5 7 4 6765 5 6 6 7 6 944 9 Solutons: Snce n s a two-dgt number, we have

More information