Lecture 7: Borel Sets and Lebesgue Measure


 Bruce Gervase Jacobs
 2 years ago
 Views:
Transcription
1 EE50: Probability Foudatios for Electrical Egieers JulyNovember 205 Lecture 7: Borel Sets ad Lebesgue Measure Lecturer: Dr. Krisha Jagaatha Scribes: Ravi Kolla, Aseem Sharma, Vishakh Hegde I this lecture, we discuss the case where the sample space is ucoutable. This case is more ivolved tha the case of a coutable sample space, maily because it is ofte ot possible to assig probabilities to all subsets of Ω. Istead, we are forced to work with a smaller σalgebra. We cosider assigig a uiform probability measure o the uit iterval. 7. Ucoutable sample spaces Cosider the experimet of pickig a real umber at radom from Ω = [0, ], such that every umber is equally likely to be picked. It is quite apparet that a simple strategy of assigig probabilities to sigleto subsets of the sample space gets ito difficulties quite quickly. Ideed, i) If we assig some positive probability to each elemetary outcome, the the probability of a evet with ifiitely may elemets, such as A = {, 2, 3, }, would become ubouded. ii) If we assig zero probability to each elemetary outcome, this aloe would ot be sufficiet to determie the probability of a ucoutable subset of Ω, such as [ 2, 3] 2. This is because probability measures are ot additive over ucoutable disjoit uios of sigletos i this case). Thus, we eed a differet approach to assig probabilities whe the sample space is ucoutable, such as Ω = [0, ]. I particular, we eed to assig probabilities directly to specific subsets of Ω. Ituitively, we would like our uiform measure µ o [0, ] to possess the followig two properties. i) µ a, b)) = µ a, b]) = µ [a, b)) = µ [a, b]) ii) Traslatioal Ivariace. That is, if A [0, ], the for ay x Ω, µ A x) = µ A) where, the set A x is defied as A x = {a + x a A, a + x } {a + x a A, a + x > } However, the followig impossibility result asserts that there is o way to cosistetly defie a uiform measure o all subsets of [0, ]. Theorem 7. Impossibility Result) There does ot exist a defiitio of a measure µ A) for all subsets of [0, ] satisfyig i) ad ii). Proof: Refer propositio.2.6 i []. Therefore, we must compromise, ad cosider a smaller σalgebra that cotais certai ice subsets of the sample space [0, ]. These ice subsets are the itervals, ad the resultig σalgebra is called the Borel σalgebra. Before defiig Borel sets, we itroduce the cocept of geeratig σalgebras from a give collectio of subsets. 7
2 72 Lecture 7: Borel Sets ad Lebesgue Measure 7.2 Geerated σalgebra ad Borel sets The σalgebra geerated by a collectio of subsets of the sample space is the smallest σalgebra that cotais the collectio. More formally, we have the followig theorem. Theorem 7.2 Let C be a arbitrary collectio of subsets of Ω, the there exists a smallest σalgebra, deoted by σ C), that cotais all elemets of C. That is, if H is ay σalgebra such that C H, the σ C) H. σ C) is called the σalgebra geerated by C. Proof: Let {F i, i I} deote the collectio of all σalgebras that cotai C. Clearly, the collectio {F i, i I} is oempty, sice it cotais at least the power set, 2 Ω. Cosider the itersectio F i. Sice the itersectio of σalgebras results i a σalgebra homework problem!) ad the itersectio cotais C, it follows that F i is a σalgebra that cotais C. Fially, if C H, the H is oe of F i s for some i I. i I Hece F i is the smallest σalgebra geerated by C. i I Ituitively, we ca thik of C as beig the collectio of subsets of Ω which are of iterest to us. The, σc) is the smallest σalgebra cotaiig all the iterestig subsets. We are ow ready to defie Borel sets. Defiitio 7.3 a) Cosider Ω = 0, ]. Let C 0 be the collectio of all ope itervals i 0, ]. The σ C 0 ), the σ  algebra geerated by C 0, is called the Borel σ  algebra. It is deoted by B 0, ]). b) A elemet of B 0, ]) is called a Borelmeasurable set, or simply a Borel set. i I Thus, every ope iterval i 0, ] is a Borel set. We ext prove that every sigleto set i 0, ] is a Borel set. Lemma 7.4 Every sigleto set {b}, 0 < b, is a Borel set, i.e., {b} B 0, ]). Proof: Cosider the collectio of sets set { b, b + } ),. By the defiitio of Borel sets, b, b + ) B 0, ]). Usig the properties of σalgebra, = = = b, b + ) c B 0, ]) b, b + ) c B 0, ]) b, b + ) ) c B 0, ]) b, b + ) B 0, ]). 7.)
3 Lecture 7: Borel Sets ad Lebesgue Measure 73 Next, we claim that {b} = b, b + ). 7.2) i.e., b is the oly elemet i b, b + ). We prove this by cotradictio. Let h be a elemet i b, b + ) other tha b. For every such h, there exists a large eough 0 such that h / b 0, b + ). 0 This implies h / b, b + ). Usig 7.) ad 7.2), thus, proves that {b} B 0, ]). As a immediate cosequece to this lemma, we see that every half ope iterval, a, b], is a Borel set. This follows from the fact that a, b] = a, b) {b}, ad the fact that a coutable uio of Borel sets is a Borel set. For the same reaso, every closed iterval, [a, b], is a Borel set. Note: Arbitrary uio of ope sets is always a ope set, but ifiite itersectios of ope sets eed ot be ope. Further readig for the ethusiastic: try Wikipedia for a start) NoBorel sets Nomeasurable sets Vitali set) BaachTarski paradox a bizzare pheomeo about cuttig up the surface of a sphere. See https: // The cardiality of the Borel σalgebra o the uit iterval) is the same as the cardiality of the reals. Thus, the Borel σalgebra is a much smaller collectio tha the power set 2 [0,]. See https: //math.dartmouth.edu/archive/m03f08/public_html/borelsetssol.pdf 7.3 Caratheodory s Extesio Theorem I this sectio, we discuss a formal procedure to defie a probability measure o a geeral measurable space Ω, F). Specifyig the probability measure for all the elemets of F directly is difficult, so we start with a smaller collectio F 0 of iterestig subsets of Ω, which eed ot be a σalgebra. We should take F 0 to be rich eough, so that the σalgebra it geerates is same as F. The we defie a fuctio P 0 : F 0 [0, ], such that it correspods to the probabilities we would like to assig to the iterestig subsets i F 0. Uder certai coditios, this fuctio P 0 ca be exteded to a legitimate probability measure o Ω, F) by usig the followig fudametal theorem from measure theory. Theorem 7.5 Caratheodory s extesio theorem) Let F 0 be a algebra of subsets of Ω, ad let F = σ F 0 ) be the σalgebra that it geerates. Suppose that P 0 is a mappig from F 0 to [0, ] that satisfies P 0 Ω) =, as well as coutable additivity o F 0. The, P 0 ca be exteded uiquely to a probability measure o Ω, F). That is, there exists a uique probability measure P o Ω, F) such that P A) = P 0 A) for all A F 0.
4 74 Lecture 7: Borel Sets ad Lebesgue Measure Proof: Refer Appedix A of [2]. We use this theorem to defie a uiform measure o 0, ], which is also called the Lebesgue measure. 7.4 The Lebesgue measure Cosider Ω = 0, ]. Let F 0 cosist of the empty set ad all sets that are fiite uios of the itervals of the form a, b]. A typical elemet of this set is of the form F = a, b ] a 2, b 2 ]... a, b ] where, 0 a < b a 2 < b 2... a < b ad N. Lemma 7.6 a) F 0 is a algebra b) F 0 is ot a σalgebra c) σ F 0 ) = B Proof: a) By defiitio, Φ F 0. Also, Φ C = 0, ] F 0. The complemet of a, b ] a 2, b 2 ] is 0, a ] b, a 2 ] b 2, ], which also belogs to F 0. Furthermore, the uio of fiitely may sets each of which are fiite uios of the itervals of the form a, b], is also a set which is the uio of fiite umber of itervals, ad thus belogs to F 0. b) To see this, ote that ] 0, + F 0 for every, but ] 0, + = 0, ) / F 0. c) First, the ull set is clearly a Borel set. Next, we have already see that every iterval of the form a, b] is a Borel set. Hece, every elemet of F 0 other tha the ull set), which is a fiite uio of such itervals, is also a Borel set. Therefore, F 0 B. This implies σ F 0 ) B. Next we show that B σ F 0 ). For ay iterval of the form a, b) i C 0, we ca write a, b) = ] ) a, b Ω. Sice every iterval of the form a, b ] F0, a coutable umber of uios of such itervals belogs to σ F 0 ). Therefore, a, b) σ F 0 ) ad cosequetly, C 0 σ F 0 ). This gives σ C 0 ) σ F 0 ). Usig the fact that σ C 0 ) = B proves the required result. For every F F 0 of the form F = a, b ] a 2, b 2 ]... a, b ], we defie a fuctio P 0 : F 0 [0, ] such that P 0 Φ) = 0 ad P 0 F ) = b i a i ).
5 Lecture 7: Borel Sets ad Lebesgue Measure 75 Note that P 0 Ω) = P 0 0, ]) =. Also, if a, b ], a 2, b 2 ],..., a, b ] are disjoit sets, the ) P 0 a i, b i ]) = P 0 a i, b i ]) = b i a i ) implyig fiite additivity of P 0. It turs out that P 0 is coutably additive o F 0 as well i.e., if a, b ], a 2, b 2 ],... are disjoit sets such that ) a i, b i ]) F 0, the P 0 a i, b i ]) = P 0 a i, b i ]) = b i a i ). The proof is otrivial ad beyod the scope of this course see [Williams] for a proof). Thus, i view of Theorem 7.5, there exists a uique probability measure P o 0, ], B) which is the same as P 0 o F 0. This uique probability measure o 0, ] is called the Lebesgue or uiform measure. The Lebesgue measure formalizes the otio of legth. This suggests that the Lebesgue measure of a sigleto should be zero. This ca be show as follows. Let b 0, ]. Usig 7.2), we write P {b}) = P b ] ), b Ω Let A = b, b]. For each, the lebesgue measure of A is P A ) = 7.3) Sice A is a decreasig sequece of ested sets, ) P {b}) =P A = lim P A ) = lim where the secod equality follows from the cotiuity of probability measures. =0 Sice ay coutable set is a coutable uio of sigletos, the probability of a coutable set is zero. For example, uder the uiform measure o 0, ], the probability of the set of ratioals is zero, sice the ratioal umbers i 0, ] form a coutable set. For Ω = 0, ], the Lebesgue measure is also a probability measure. For other itervals for example Ω = 0, 2]), it will oly be a fiite measure, which ca be ormalized as appropriate to obtai a uiform probability measure. Defiitio 7.7 Let Ω, F, P) be a probability space. PA) =. A evet A is said to occur almost surely a.s) if Cautio: PA) = does ot mea A = Ω.
6 76 Lecture 7: Borel Sets ad Lebesgue Measure Lebesgue Measure of the Cator set: Cosider the cator set K. It is created by repeatedly removig the ope middle thirds of a set of lie segmets. Cosider its complemet. It cotais coutable umber of disjoit itervals. Hece we have: PK c ) = = =. Therefore PK) = 0. It is very iterestig to ote that though the Cator set is equicardial with 0, ], its Lebesgue measure is equal to 0 while the Lebesgue measure of 0, ] is equal to. We ow exted the defiitio of Lebesgue measure o [0, ] to the real lie, R. We first look at the defiitio of a Borel set o R. This ca be doe i several ways, as show below. Defiitio 7.8 Borel sets o R: Let C be a collectio of ope itervals i R. The BR) = σc) is the Borel set o R. Let D be a collectio of semiifiite itervals {, x]; x R}, the σd) = BR). A R is said to be a Borel set o R, if A, + ] is a Borel set o, + ] Z. Exercise: Verify that the three statemets are equivalet defiitios of Borel sets o R. Defiitio 7.9 Lebesgue measure of A R: λa) = P A, + ]) = Theorem 7.0 R, BR), λ) is a ifiite measure space. Proof: We eed to prove followig: λr) = λφ) = 0 The coutable additivity property We see that P R, + ]) =, I Hece we have λr) = = = Now cosider Φ, + ]. This is a ull set for all. Hece we have, P Φ, + ]) = 0, I which implies, λφ) = P Φ, + ]) = 0 =
7 Lecture 7: Borel Sets ad Lebesgue Measure 77 We ow eed to prove the coutable additivity property. For this we cosider A i BR) such that the sequece A, A 2,..., A,... are arbitrary pairwise disjoit sets i BR). Therefore we obtai, λ A i ) = = = = P = = A i, + ]) P A i, + ]) P A i, + ]) The secod equality above comes from the fact that the probability measure has coutable additivity property. The last equality above comes from the fact that the summatios ca be iterchaged from Fubii s theorem). We also have the followig: λa i ) = = P A i, + ]) We ow immediately see that Hece proved. λ A i ) = λa i ) 7.5 Exercises. Let F be a σalgebra correspodig to a sample space Ω. Let H be a subset of Ω that does ot belog to F. Cosider the collectio G of all sets of the form H A) H c B), where A ad B F. a) Show that H A G. b) Show that G is a σalgebra. 2. Show that C = σc) iff C is a σalgebra. 3. Let C ad D be two collectios of subsets of Ω such that C D. Prove that σc) σd). 4. Prove that the followig subsets of 0, ] are Borelmeasurable. a) ay coutable set b) the set of irratioal umbers c) the Cator set Hit: rather tha defiig it i terms of terary expasios, it s easier to use the equivalet defiitio of the Cator set that ivolves sequetially removig the middlethird ope itervals; see Wikipedia for example). d) The set of umbers i 0, ] whose decimal expasio does ot cotai Let B deote the Borel σalgebra as defied i class. Let C c deote the set of all closed itervals cotaied i 0, ]. Show that σc c ) = B. I other words, we could have very well defied the Borel σalgebra as beig geerated by closed itervals, rather tha ope itervals.
8 78 Lecture 7: Borel Sets ad Lebesgue Measure 6. Let Ω = [0, ], ad let F 3 cosist of all coutable subsets of Ω, ad all subsets of Ω havig a coutable complemet. It ca be show that F 3 is a σalgebra Refer Lecture 4, Exercises, 6d)). Let us defie PA) = 0 if A coutable, ad PA) = if A has a coutable complemet. Is Ω, F 3, P) a legitimate probability space? 7. We have see i 4c) that the Cator set is Borelmeasurable. Show that the Cator set has zero Lebesgue measure. Thus, although the Cator set ca be put ito a bijectio with [0, ], it has zero Lebesgue measure! Refereces [] Rosethal, J. S. 2006). A first look at rigorous probability theory Vol. 2). Sigapore: World Scietific. [2] Williams, D. 99). Probability with martigales. Cambridge uiversity press.
Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the BruMikowski iequality for boxes. Today we ll go over the
More informationx(x 1)(x 2)... (x k + 1) = [x] k n+m 1
1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationMARTINGALES AND A BASIC APPLICATION
MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measuretheoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More information{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers
. Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More informationOverview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability
More informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationSequences II. Chapter 3. 3.1 Convergent Sequences
Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,
More information1 Set Theory and Functions
Set Theory ad Fuctios. Basic De itios ad Notatio A set A is a collectio of objects of ay kid. We write a A to idicate that a is a elemet of A: We express this as a is cotaied i A. We write A B if every
More informationMetric, Normed, and Topological Spaces
Chapter 13 Metric, Normed, ad Topological Spaces A metric space is a set X that has a otio of the distace d(x, y) betwee every pair of poits x, y X. A fudametal example is R with the absolutevalue metric
More informationTHE ABRACADABRA PROBLEM
THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationLecture 5: Span, linear independence, bases, and dimension
Lecture 5: Spa, liear idepedece, bases, ad dimesio Travis Schedler Thurs, Sep 23, 2010 (versio: 9/21 9:55 PM) 1 Motivatio Motivatio To uderstad what it meas that R has dimesio oe, R 2 dimesio 2, etc.;
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real oegative umber R, called the radius
More information3 Basic Definitions of Probability Theory
3 Basic Defiitios of Probability Theory 3defprob.tex: Feb 10, 2003 Classical probability Frequecy probability axiomatic probability Historical developemet: Classical Frequecy Axiomatic The Axiomatic defiitio
More information1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : pvalue
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More informationThe Field Q of Rational Numbers
Chapter 3 The Field Q of Ratioal Numbers I this chapter we are goig to costruct the ratioal umber from the itegers. Historically, the positive ratioal umbers came first: the Babyloias, Egyptias ad Grees
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationEngineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51
Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log
More informationThe Euler Totient, the Möbius and the Divisor Functions
The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 007 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a uow mea µ = E(X) of a distributio by
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More informationPART TWO. Measure, Integration, and Differentiation
PART TWO Measure, Itegratio, ad Differetiatio Émile FélixÉdouardJusti Borel (1871 1956 Émile Borel was bor at SaitAffrique, Frace, o Jauary 7, 1871, the third child of Hooré Borel, a Protestat miister,
More informationLecture 4: Cheeger s Inequality
Spectral Graph Theory ad Applicatios WS 0/0 Lecture 4: Cheeger s Iequality Lecturer: Thomas Sauerwald & He Su Statemet of Cheeger s Iequality I this lecture we assume for simplicity that G is a dregular
More informationApproximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.
More informationNOTES ON PROBABILITY Greg Lawler Last Updated: March 21, 2016
NOTES ON PROBBILITY Greg Lawler Last Updated: March 21, 2016 Overview This is a itroductio to the mathematical foudatios of probability theory. It is iteded as a supplemet or followup to a graduate course
More informationA Faster ClauseShortening Algorithm for SAT with No Restriction on Clause Length
Joural o Satisfiability, Boolea Modelig ad Computatio 1 2005) 4960 A Faster ClauseShorteig Algorithm for SAT with No Restrictio o Clause Legth Evgey Datsi Alexader Wolpert Departmet of Computer Sciece
More informationChapter 7  Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:
Chapter 7  Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationFactors of sums of powers of binomial coefficients
ACTA ARITHMETICA LXXXVI.1 (1998) Factors of sums of powers of biomial coefficiets by Neil J. Cali (Clemso, S.C.) Dedicated to the memory of Paul Erdős 1. Itroductio. It is well ow that if ( ) a f,a = the
More informationDistributions of Order Statistics
Chapter 2 Distributios of Order Statistics We give some importat formulae for distributios of order statistics. For example, where F k: (x)=p{x k, x} = I F(x) (k, k + 1), I x (a,b)= 1 x t a 1 (1 t) b 1
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationPermutations, the Parity Theorem, and Determinants
1 Permutatios, the Parity Theorem, ad Determiats Joh A. Guber Departmet of Electrical ad Computer Egieerig Uiversity of Wiscosi Madiso Cotets 1 What is a Permutatio 1 2 Cycles 2 2.1 Traspositios 4 3 Orbits
More informationLinear Algebra II. 4 Determinants. Notes 4 1st November Definition of determinant
MTH6140 Liear Algebra II Notes 4 1st November 2010 4 Determiats The determiat is a fuctio defied o square matrices; its value is a scalar. It has some very importat properties: perhaps most importat is
More informationNotes on exponential generating functions and structures.
Notes o expoetial geeratig fuctios ad structures. 1. The cocept of a structure. Cosider the followig coutig problems: (1) to fid for each the umber of partitios of a elemet set, (2) to fid for each the
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More informationOur aim is to show that under reasonable assumptions a given 2πperiodic function f can be represented as convergent series
8 Fourier Series Our aim is to show that uder reasoable assumptios a give periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More informationChapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity
More informationIrreducible polynomials with consecutive zero coefficients
Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem
More information3. Covariance and Correlation
Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed MultiEvent Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed MultiEvet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More informationWHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?
WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This
More informationON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE
Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE by Maria E Gageoea ad Silvia Moldoveau
More information4. Trees. 4.1 Basics. Definition: A graph having no cycles is said to be acyclic. A forest is an acyclic graph.
4. Trees Oe of the importat classes of graphs is the trees. The importace of trees is evidet from their applicatios i various areas, especially theoretical computer sciece ad molecular evolutio. 4.1 Basics
More informationAnnuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.
Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio  Israel Istitute of Techology, 3000, Haifa, Israel I memory
More information1.3 Binomial Coefficients
18 CHAPTER 1. COUNTING 1. Biomial Coefficiets I this sectio, we will explore various properties of biomial coefficiets. Pascal s Triagle Table 1 cotais the values of the biomial coefficiets ( ) for 0to
More informationAnalysis Notes (only a draft, and the first one!)
Aalysis Notes (oly a draft, ad the first oe!) Ali Nesi Mathematics Departmet Istabul Bilgi Uiversity Kuştepe Şişli Istabul Turkey aesi@bilgi.edu.tr Jue 22, 2004 2 Cotets 1 Prelimiaries 9 1.1 Biary Operatio...........................
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationON THE EDGEBANDWIDTH OF GRAPH PRODUCTS
ON THE EDGEBANDWIDTH OF GRAPH PRODUCTS JÓZSEF BALOGH, DHRUV MUBAYI, AND ANDRÁS PLUHÁR Abstract The edgebadwidth of a graph G is the badwidth of the lie graph of G We show asymptotically tight bouds o
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 KolmogorovSmirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More information4.3. The Integral and Comparison Tests
4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece
More informationThe following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles
The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationChapter One BASIC MATHEMATICAL TOOLS
Chapter Oe BAIC MATHEMATICAL TOOL As the reader will see, the study of the time value of moey ivolves substatial use of variables ad umbers that are raised to a power. The power to which a variable is
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationSection 1.6: Proof by Mathematical Induction
Sectio.6 Proof by Iductio Sectio.6: Proof by Mathematical Iductio Purpose of Sectio: To itroduce the Priciple of Mathematical Iductio, both weak ad the strog versios, ad show how certai types of theorems
More informationPlugin martingales for testing exchangeability online
Plugi martigales for testig exchageability olie Valetia Fedorova, Alex Gammerma, Ilia Nouretdiov, ad Vladimir Vovk Computer Learig Research Cetre Royal Holloway, Uiversity of Lodo, UK {valetia,ilia,alex,vovk}@cs.rhul.ac.uk
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationJournal of Combinatorial Theory, Series A
Joural of Combiatorial Theory, Series A 118 011 319 345 Cotets lists available at ScieceDirect Joural of Combiatorial Theory, Series A www.elsevier.com/locate/jcta Geeratig all subsets of a fiite set with
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More information7. Sample Covariance and Correlation
1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y
More informationSampling Distribution And Central Limit Theorem
() Samplig Distributio & Cetral Limit Samplig Distributio Ad Cetral Limit Samplig distributio of the sample mea If we sample a umber of samples (say k samples where k is very large umber) each of size,
More informationTAYLOR SERIES, POWER SERIES
TAYLOR SERIES, POWER SERIES The followig represets a (icomplete) collectio of thigs that we covered o the subject of Taylor series ad power series. Warig. Be prepared to prove ay of these thigs durig the
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More informationarxiv:1012.1336v2 [cs.cc] 8 Dec 2010
Uary SubsetSum is i Logspace arxiv:1012.1336v2 [cs.cc] 8 Dec 2010 1 Itroductio Daiel M. Kae December 9, 2010 I this paper we cosider the Uary SubsetSum problem which is defied as follows: Give itegers
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationPart  I. Mathematics
Part  I Mathematics CHAPTER Set Theory. Objectives. Itroductio. Set Cocept.. Sets ad Elemets. Subset.. Proper ad Improper Subsets.. Equality of Sets.. Trasitivity of Set Iclusio.4 Uiversal Set.5 Complemet
More informationRamseytype theorems with forbidden subgraphs
Ramseytype theorems with forbidde subgraphs Noga Alo Jáos Pach József Solymosi Abstract A graph is called Hfree if it cotais o iduced copy of H. We discuss the followig questio raised by Erdős ad Hajal.
More informationCHAPTER 7: Central Limit Theorem: CLT for Averages (Means)
CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:
More informationDiscrete Random Variables and Probability Distributions. Random Variables. Chapter 3 3.1
UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig
More informationRANDOM GRAPHS WITH FORBIDDEN VERTEX DEGREES
RANDOM GRAPHS WITH FORBIDDEN VERTEX DEGREES GEOFFREY GRIMMETT AND SVANTE JANSON Abstract. We study the radom graph G,λ/ coditioed o the evet that all vertex degrees lie i some give subset S of the oegative
More informationTHE HEIGHT OF qbinary SEARCH TREES
THE HEIGHT OF qbinary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average
More informationINFINITE SERIES KEITH CONRAD
INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal
More informationProject Deliverables. CS 361, Lecture 28. Outline. Project Deliverables. Administrative. Project Comments
Project Deliverables CS 361, Lecture 28 Jared Saia Uiversity of New Mexico Each Group should tur i oe group project cosistig of: About 612 pages of text (ca be loger with appedix) 612 figures (please
More informationLECTURE 13: Crossvalidation
LECTURE 3: Crossvalidatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Threeway data partitioi Itroductio to Patter Aalysis Ricardo GutierrezOsua Texas A&M
More informationTHE UNLIKELY UNION OF PARTITIONS AND DIVISORS
THE UNLIKELY UNION OF PARTITIONS AND DIVISORS Abdulkadir Hasse, Thomas J. Osler, Mathematics Departmet ad Tirupathi R. Chadrupatla, Mechaical Egieerig Rowa Uiversity Glassboro, NJ 828 I the multiplicative
More information4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then
SECTION 2.6 THE RATIO TEST 79 2.6. THE RATIO TEST We ow kow how to hadle series which we ca itegrate (the Itegral Test), ad series which are similar to geometric or pseries (the Compariso Test), but of
More informationEntropy of bicapacities
Etropy of bicapacities Iva Kojadiovic LINA CNRS FRE 2729 Site école polytechique de l uiv. de Nates Rue Christia Pauc 44306 Nates, Frace iva.kojadiovic@uivates.fr JeaLuc Marichal Applied Mathematics
More informationKey Ideas Section 81: Overview hypothesis testing Hypothesis Hypothesis Test Section 82: Basics of Hypothesis Testing Null Hypothesis
Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, Pvalue Type I Error, Type II Error, Sigificace Level, Power Sectio 81: Overview Cofidece Itervals (Chapter 7) are
More informationCME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8
CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 8 GENE H GOLUB 1 Positive Defiite Matrices A matrix A is positive defiite if x Ax > 0 for all ozero x A positive defiite matrix has real ad positive
More informationSequences, Series and Convergence with the TI 92. Roger G. Brown Monash University
Sequeces, Series ad Covergece with the TI 92. Roger G. Brow Moash Uiversity email: rgbrow@deaki.edu.au Itroductio. Studets erollig i calculus at Moash Uiversity, like may other calculus courses, are itroduced
More informationClass Meeting # 16: The Fourier Transform on R n
MATH 18.152 COUSE NOTES  CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,
More informationNUMBERS COMMON TO TWO POLYGONAL SEQUENCES
NUMBERS COMMON TO TWO POLYGONAL SEQUENCES DIANNE SMITH LUCAS Chia Lake, Califoria a iteger, The polygoal sequece (or sequeces of polygoal umbers) of order r (where r is r > 3) may be defied recursively
More informationVladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT
Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee
More information