Topics in Probability Theory and Stochastic Processes Steven R. Dunbar. The Weak Law of Large Numbers

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Topics in Probability Theory and Stochastic Processes Steven R. Dunbar. The Weak Law of Large Numbers"

Transcription

1 Steve R. Dubar Departmet o Mathematics 203 Avery Hall Uiversity o Nebrasa-Licol Licol, NE Voice: Fax: Topics i Probability Theory ad Stochastic Processes Steve R. Dubar The Wea Law o Large Numbers Ratig Mathematicias Oly: prologed scees o itese rigor. 1

2 Questio o the Day Cosider a air (p = 1/2 = q) coi tossig game carried out or 1000 tosses. Explai i a setece what the law o averages says about the outcomes o this game. Be as precise as possible. Key Cocepts 1. Marov s Iequality: Let X be a radom variable taig oly oegative values. The or each a > 0 P [X > a] E [X] /a; 2. Chebyshev s Iequality: Let X be a radom variable. The or a > 0 P [ X E [X] a] Var [X] a 2 3. Wea Law o Large Numbers: For ɛ > 0 [ ] S P p > ɛ 0 as ad the covergece is uiorm i p. 4. Let be a real uctio that is deied ad cotiuous o the iterval [0, 1]. The ( ) ( ) sup (x) x (1 x) 0 as. =0 2

3 Vocabulary 1. The Wea Law o Large Numbers says that or ɛ > 0 [ ] S P p > ɛ 0 as ad the covergece is uiorm i p. 2. The polyomials B, (t) = are called the Berstei polyomials. ( ) x (1 x) Mathematical Ideas Proo o the Wea Law Usig Chebyshev s Iequality Propositio 1 (Marov s Iequality). Let X be a radom variable taig oly o-egative values. The or each a > 0 Proo. P [X a] E [X] /a. P [X a] = E [I X a ] = dp [] X a x dp [] a 1 a E [X] 3

4 Propositio 2 (Chebyshev s Iequality). Let X be a radom variable. The or a > 0 Var [X] P [ X E [X] a]. a 2 Proo. This immediately ollows rom Marov s iequality applied to the oegative radom variable (X E [X]) 2. Theorem 3 (Wea Law o Large Numbers). For ɛ > 0 [ ] S P p > ɛ 0 as ad the covergece is uiorm i p. Remar. The otatio P [] idicates that we are cosiderig a amily o probability measures o the sample space Ω. The Wea Law establishes the covergece o the sequece o measures i a particular way. Proo. The variace o the radom variable S is p(1 p). Rewrite the probability as the equivalet evet: [ ] S P p > ɛ = P [ S p > ɛ]. By Chebyshev s iequality P [ S p > ɛ] Var [S ] (ɛ) 2 = Sice p(1 p) 1/4, the proo is complete. Remar. This iequality demostrates that [ ] S P p > ɛ = O(1/) uiormly i p. p(1 p) ɛ 2 1. Remar. Jacob Beroulli origially proved the Wea Law o Large Numbers i 1713 or the special case whe the X i are biomial radom variables. Beroulli had to create a igeious proo to establish the result, sice Chebyshev s iequality was ot ow at the time. The theorem the 4

5 became ow as Beroulli s Theorem. Simeo Poisso proved a geeralizatio o Beroulli s biomial Wea Law ad irst called it the Law o Large Numbers. I 1929 the Russia mathematicia Alesadr Khichi proved the geeral orm o the Wea Law o Large Numbers preseted here. May other versios o the Wea Law are ow, with hypotheses that do ot require such striget requiremets as beig idetically distributed, ad havig iite variace. Remar. Aother proo o the Wea Law o Large Numbers usig momet geeratig uctios is i Mathematical Fiace/Cetral Limit Theorem Berstei s Proo o the Weierstrass Approximatio Theorem Theorem 4. Let be a real uctio deied ad cotiuous o the iterval [0, 1]. The ( ) ( ) sup (x) x (1 x) 0 as. =0 Proo. 1. Fix ɛ > 0. Sice cotiuous o the compact iterval [0, 1] it is uiormly cotiuous o [0, 1]. Thereore there is a η > 0 such that (x) (y) < ɛ i x y < η. 2. The expectatio E [(S /)] ca be expressed as a polyomial i p: [ E ( )] S = =0 ( ) P [S = ] = =0 ( ) ( ) p (1 p). 3. By the Wea Law o Large Numbers, or the give ɛ > 0, there is a 0 such that [ ] S P p > η < ɛ. 4. E [ ( ) S (p)] = =0 5 ( ( ) ) (p) P [S = ].

6 5. Apply the triagle iequality to the right had side ad express i terms o two summatios: ( ( ) ) (p) P [S = ] + p η ( ( ) ) + (p) P [S = ] p >η Note the secod applicatio o the triagle iequality o the secod summatio. 6. Now estimate the terms: ɛp [S = ] + p η p >η 2 sup (x) P [S = ] 7. Fially, do the additio over the idividual values o the probabilities over sigle values to re-write them as probabilities over evets: [ ] [ ] S = ɛp p η S + 2 sup (x) P p > η 8. Now apply the Wea Law to the secod term to see that: ( ) [ E S (p)] < ɛ + 2ɛ sup (x). This shows that E [ ( S ) (p) ] ca be made arbitrarily small, uiormly with respect to p, by picig suicietly large. Remar. The polyomials B, (t) = ( ) x (1 x) are called the Berstei polyomials. The Berstei polyomials have several useul properties: 6

7 1. B i, (t) = B i, (1 t) 2. B i, (t) 0 3. i=0 B i,(t) = 1 or 0 t 1. Corollary 1. A polyomial o degree uiormly approximatig the cotiuous uctio (x) o the iterval [a, b] is Sources =0 ( a + (b a) ) ( ) ( x a b a ) ( ) b x b a This sectio is adapted rom: Heads or Tails, by Emmauel Lesige, Studet Mathematical Library Volume 28, America Mathematical Society, Providece, 2005, Chapter 5, [3]. Problems to Wor or Uderstadig 1. Suppose X is a cotiuous radom variable with mea ad variace both equal to 20. What ca be said about P [0 X 40]? 2. Suppose X is a expoetially distributed radom variable with mea E [X] = 1. For x = 0.5, 1, ad 2, compare P [X x] with the Marov iequality boud. 3. Suppose X is a Beroulli radom variable with P [X = 1] = p ad P [X = 0] = 1 p = q. Compare P [X 1] with the Marov iequality boud. 4. Let X 1, X 2,..., X 10 be idepedet Poisso radom variables with mea 1. First use the Marov Iequality to get a boud o P [X X 10 > 15]. Next id the exact probability that P [X X 10 > 15] usig that the act that the sum o idepedet Poisso radom variables with parameters λ 1, λ 2 is agai Poisso with parameter λ 1 + λ 2. 7

8 5. Cosider a air (p = 1/2 = q) coi tossig game carried out or = 100 tosses. Calculate the exact value [ ] S P p > 1/10 ad compare it to the estimates i the proo o the Wea Law o Large Numbers. 6. Calculate the Berstei polyomial approximatio o si(πx) o degree 1, 2, ad 3 ad plot the graphs o si(πx) ad the approximatios. 7. Calculate the Berstei polyomial approximatio o cos(πx) o degree 1, 2, ad 3 ad plot the graphs o cos(πx) ad the approximatios. 8. Calculate the Berstei polyomial approximatio o exp(πx) o degree 1, 2, ad 3 ad plot the graphs o exp(πx) ad the approximatios. Readig Suggestio: Reereces [1] Leo Breima. Probability. SIAM, [2] William Feller. A Itroductio to Probability Theory ad Its Applicatios, Volume I, volume I. Joh Wiley ad Sos, third editio, QA 273 F3712. [3] Emmauel Lesige. Heads or Tails: A Itroductio to Limit Theorems i Probability, volume 28 o Studet Mathematical Library. America Mathematical Society,

9 Outside Readigs ad Lis: 1. Virtual Laboratories i Probability ad Statistics / Biomial 2. Weisstei, Eric W. Wea Law o Large Numbers. From MathWorld A Wolram Web Resource. Wea Law o Large Numbers. 3. Wiipedia, Wea Law o Large Numbers I chec all the iormatio o each page or correctess ad typographical errors. Nevertheless, some errors may occur ad I would be grateul i you would alert me to such errors. I mae every reasoable eort to preset curret ad accurate iormatio or public use, however I do ot guaratee the accuracy or timeliess o iormatio o this website. Your use o the iormatio rom this website is strictly volutary ad at your ris. I have checed the lis to exteral sites or useuless. Lis to exteral websites are provided as a coveiece. I do ot edorse, cotrol, moitor, or guaratee the iormatio cotaied i ay exteral website. I do t guaratee that the lis are active at all times. Use the lis here with the same cautio as you would all iormatio o the Iteret. This website relects the thoughts, iterests ad opiios o its author. They do ot explicitly represet oicial positios or policies o my employer. Iormatio o this website is subject to chage without otice. Steve Dubar s Home Page, to Steve Dubar, sdubar1 at ul dot edu Last modiied: Processed rom L A TEX source o May 25,

Topics in Probability Theory and Stochastic Processes Steven R. Dunbar. Binomial Distribution

Topics in Probability Theory and Stochastic Processes Steven R. Dunbar. Binomial Distribution Steven R. Dunbar Department of Mathematics 203 Avery Hall University of Nebrasa-Lincoln Lincoln, NE 68588-0130 http://www.math.unl.edu Voice: 402-472-3731 Fax: 402-472-8466 Topics in Probability Theory

More information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In 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 information

Sequences II. Chapter 3. 3.1 Convergent Sequences

Sequences 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 information

A probabilistic proof of a binomial identity

A 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 information

Properties of MLE: consistency, asymptotic normality. Fisher information.

Properties 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 information

Module 4: Mathematical Induction

Module 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 information

A Recursive Formula for Moments of a Binomial Distribution

A Recursive Formula for Moments of a Binomial Distribution A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,

More information

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006 Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam

More information

3. Covariance and Correlation

3. 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 information

Sequences and Series

Sequences 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 information

4.1 Sigma Notation and Riemann Sums

4.1 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 information

Overview of some probability distributions.

Overview 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 information

Riemann Sums y = f (x)

Riemann Sums y = f (x) Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, o-egative fuctio o the closed iterval [a, b] Fid

More information

4 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

4 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 p-series (the Compariso Test), but of

More information

Mathematicians have been fascinated by the majestic simplicity of the Fibonacci

Mathematicians have been fascinated by the majestic simplicity of the Fibonacci Joh Holde Tutoa3000@aol.com Ivertig the iboacci Sequece Mathematicias have bee fasciated by the majestic simplicity of the iboacci Sequece for ceturies. It starts as a simple,,, 3, 5, 8,3,... computed

More information

ORDERS OF GROWTH KEITH CONRAD

ORDERS OF GROWTH KEITH CONRAD ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus

More information

8 The Poisson Distribution

8 The Poisson Distribution 8 The Poisso Distributio Let X biomial, p ). Recall that this meas that X has pmf ) p,p k) p k k p ) k for k 0,,...,. ) Agai, thik of X as the umber of successes i a series of idepedet experimets, each

More information

Measurable Functions

Measurable Functions Measurable Fuctios Dug Le 1 1 Defiitio It is ecessary to determie the class of fuctios that will be cosidered for the Lebesgue itegratio. We wat to guaratee that the sets which arise whe workig with these

More information

Matrix Transforms of A-statistically Convergent Sequences with Speed

Matrix Transforms of A-statistically Convergent Sequences with Speed Filomat 27:8 2013, 1385 1392 DOI 10.2298/FIL1308385 Published by Faculty of Scieces ad Mathematics, Uiversity of Niš, Serbia vailable at: http://www.pmf.i.ac.rs/filomat Matrix Trasforms of -statistically

More information

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients 652 (12-26) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

if A S, then X \ A S, and if (A n ) n is a sequence of sets in S, then n A n S,

if A S, then X \ A S, and if (A n ) n is a sequence of sets in S, then n A n S, Lecture 5: Borel Sets Topologically, the Borel sets i a topological space are the σ-algebra geerated by the ope sets. Oe ca build up the Borel sets from the ope sets by iteratig the operatios of complemetatio

More information

Simulation and Monte Carlo integration

Simulation and Monte Carlo integration Chapter 3 Simulatio ad Mote Carlo itegratio I this chapter we itroduce the cocept of geeratig observatios from a specified distributio or sample, which is ofte called Mote Carlo geeratio. The ame of Mote

More information

Incremental calculation of weighted mean and variance

Incremental 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 information

Continuous Random Variables: Joint PDFs, Conditioning, Expectation and Independence

Continuous Random Variables: Joint PDFs, Conditioning, Expectation and Independence Cotiuous Radom Variables: Joit DFs, Coditioig, xpectatio ad Idepedece Berli Che Departmet o Computer ciece & Iormatio gieerig Natioal Taiwa Normal Uiversit Reerece: - D.. Bertsekas, J. N. Tsitsiklis, Itroductio

More information

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, 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 information

Asymptotic Growth of Functions

Asymptotic 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 information

1 Computing the Standard Deviation of Sample Means

1 Computing the Standard Deviation of Sample Means Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

More information

Winter Camp 2012 Sequences Alexander Remorov. Sequences. Alexander Remorov

Winter Camp 2012 Sequences Alexander Remorov. Sequences. Alexander Remorov Witer Camp 202 Sequeces Alexader Remorov Sequeces Alexader Remorov alexaderrem@gmail.com Warm-up Problem : Give a positive iteger, cosider a sequece of real umbers a 0, a,..., a defied as a 0 = 2 ad =

More information

THE HEIGHT OF q-binary SEARCH TREES

THE HEIGHT OF q-binary SEARCH TREES THE HEIGHT OF q-binary 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 information

3.2 Introduction to Infinite Series

3.2 Introduction to Infinite Series 3.2 Itroductio to Ifiite Series May of our ifiite sequeces, for the remaider of the course, will be defied by sums. For example, the sequece S m := 2. () is defied by a sum. Its terms (partial sums) are

More information

Department of Computer Science, University of Otago

Department of Computer Science, University of Otago Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

More information

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method Chapter 6: Variace, the law of large umbers ad the Mote-Carlo 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 information

Output Analysis (2, Chapters 10 &11 Law)

Output Analysis (2, Chapters 10 &11 Law) B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should

More information

A RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY

A RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY J. Appl. Prob. 45, 060 070 2008 Prited i Eglad Applied Probability Trust 2008 A RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY MARK BROWN, The City College of New York EROL A. PEKÖZ, Bosto Uiversity SHELDON

More information

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means)

CHAPTER 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 information

Hypothesis testing. Null and alternative hypotheses

Hypothesis 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 information

Example 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).

Example 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 information

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:

Chapter 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 information

1 Introduction to reducing variance in Monte Carlo simulations

1 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 information

1. C. The formula for the confidence interval for a population mean is: x t, which was

1. 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 : p-value

More information

Confidence Intervals for the Mean of Non-normal Data Class 23, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Confidence Intervals for the Mean of Non-normal Data Class 23, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Cofidece Itervals for the Mea of No-ormal Data Class 23, 8.05, Sprig 204 Jeremy Orloff ad Joatha Bloom Learig Goals. Be able to derive the formula for coservative ormal cofidece itervals for the proportio

More information

8.5 Alternating infinite series

8.5 Alternating infinite series 65 8.5 Alteratig ifiite series I the previous two sectios we cosidered oly series with positive terms. I this sectio we cosider series with both positive ad egative terms which alterate: positive, egative,

More information

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)

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 Bru-Mikowski iequality for boxes. Today we ll go over the

More information

0.7 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

0.7 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 Kolmogorov-Smirov 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 information

MIDTERM EXAM - MATH 563, SPRING 2016

MIDTERM EXAM - MATH 563, SPRING 2016 MIDTERM EXAM - MATH 563, SPRING 2016 NAME: SOLUTION Exam rules: There are 5 problems o this exam. You must show all work to receive credit, state ay theorems ad defiitios clearly. The istructor will NOT

More information

NPTEL STRUCTURAL RELIABILITY

NPTEL STRUCTURAL RELIABILITY NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics

More information

1 Notes on Little s Law (l = λw)

1 Notes on Little s Law (l = λw) Copyright c 29 by Karl Sigma Notes o Little s Law (l λw) We cosider here a famous ad very useful law i queueig theory called Little s Law, also kow as l λw, which asserts that the time average umber of

More information

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.

Annuities 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 information

SAMPLE 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. (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

Topics in Probability Theory and Stochastic Processes Steven R. Dunbar. Stirling s Formula derived from the Poisson Distribution

Topics in Probability Theory and Stochastic Processes Steven R. Dunbar. Stirling s Formula derived from the Poisson Distribution Steven R. Dunbar Department of Mathematics 203 Avery Hall University of Nebrasa-Lincoln Lincoln, NE 68588-0130 http://www.math.unl.edu Voice: 402-472-3731 Fax: 402-472-8466 Topics in Probability Theory

More information

An Efficient Polynomial Approximation of the Normal Distribution Function & Its Inverse Function

An Efficient Polynomial Approximation of the Normal Distribution Function & Its Inverse Function A Efficiet Polyomial Approximatio of the Normal Distributio Fuctio & Its Iverse Fuctio Wisto A. Richards, 1 Robi Atoie, * 1 Asho Sahai, ad 3 M. Raghuadh Acharya 1 Departmet of Mathematics & Computer Sciece;

More information

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.

More information

I. Chi-squared Distributions

I. Chi-squared Distributions 1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

More information

Section 9.2 Series and Convergence

Section 9.2 Series and Convergence Sectio 9. Series ad Covergece Goals of Chapter 9 Approximate Pi Prove ifiite series are aother importat applicatio of limits, derivatives, approximatio, slope, ad cocavity of fuctios. Fid challegig atiderivatives

More information

Derivation of the Poisson distribution

Derivation of the Poisson distribution Gle Cowa RHUL Physics 1 December, 29 Derivatio of the Poisso distributio I this ote we derive the fuctioal form of the Poisso distributio ad ivestigate some of its properties. Cosider a time t i which

More information

Arithmetic Sequences

Arithmetic Sequences . Arithmetic Sequeces Essetial Questio How ca you use a arithmetic sequece to describe a patter? A arithmetic sequece is a ordered list of umbers i which the differece betwee each pair of cosecutive terms,

More information

1 The Binomial Theorem: Another Approach

1 The Binomial Theorem: Another Approach The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets

More information

Sequences, Series and Convergence with the TI 92. Roger G. Brown Monash University

Sequences, 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 information

Estimating the Mean and Variance of a Normal Distribution

Estimating the Mean and Variance of a Normal Distribution Estimatig the Mea ad Variace of a Normal Distributio Learig Objectives After completig this module, the studet will be able to eplai the value of repeatig eperimets eplai the role of the law of large umbers

More information

The geometric series and the ratio test

The geometric series and the ratio test The geometric series ad the ratio test Today we are goig to develop aother test for covergece based o the iterplay betwee the it compariso test we developed last time ad the geometric series. A ote about

More information

Approximating the Sum of a Convergent Series

Approximating the Sum of a Convergent Series Approximatig the Sum of a Coverget Series Larry Riddle Ages Scott College Decatur, GA 30030 lriddle@agesscott.edu The BC Calculus Course Descriptio metios how techology ca be used to explore covergece

More information

Linear Algebra II. Notes 6 25th November 2010

Linear Algebra II. Notes 6 25th November 2010 MTH6140 Liear Algebra II Notes 6 25th November 2010 6 Quadratic forms A lot of applicatios of mathematics ivolve dealig with quadratic forms: you meet them i statistics (aalysis of variace) ad mechaics

More information

Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis

Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, P-value Type I Error, Type II Error, Sigificace Level, Power Sectio 8-1: Overview Cofidece Itervals (Chapter 7) are

More information

Convexity, Inequalities, and Norms

Convexity, 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 information

3 Basic Definitions of Probability Theory

3 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 information

Empirical Distributions

Empirical Distributions Emirical Distributios A emirical distributio is oe for which each ossible evet is assiged a robability derived from exerimetal observatio. It is assumed that the evets are ideedet ad the sum of the robabilities

More information

Binet Formulas for Recursive Integer Sequences

Binet Formulas for Recursive Integer Sequences Biet Formulas for Recursive Iteger Sequeces Homer W. Austi Jatha W. Austi Abstract May iteger sequeces are recursive sequeces ad ca be defied either recursively or explicitly by use of Biet-type formulas.

More information

Lecture 7: Borel Sets and Lebesgue Measure

Lecture 7: Borel Sets and Lebesgue Measure EE50: Probability Foudatios for Electrical Egieers July-November 205 Lecture 7: Borel Sets ad Lebesgue Measure Lecturer: Dr. Krisha Jagaatha Scribes: Ravi Kolla, Aseem Sharma, Vishakh Hegde I this lecture,

More information

Section 11.3: The Integral Test

Section 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 information

2.7 Sequences, Sequences of Sets

2.7 Sequences, Sequences of Sets 2.7. SEQUENCES, SEQUENCES OF SETS 67 2.7 Sequeces, Sequeces of Sets 2.7.1 Sequeces Defiitio 190 (sequece Let S be some set. 1. A sequece i S is a fuctio f : K S where K = { N : 0 for some 0 N}. 2. For

More information

2.3. GEOMETRIC SERIES

2.3. GEOMETRIC SERIES 6 CHAPTER INFINITE SERIES GEOMETRIC SERIES Oe of the most importat types of ifiite series are geometric series A geometric series is simply the sum of a geometric sequece, Fortuately, geometric series

More information

Contents. 7 Sequences and Series. 7.1 Sequences and Convergence. Calculus II (part 3): Sequences and Series (by Evan Dummit, 2015, v. 2.

Contents. 7 Sequences and Series. 7.1 Sequences and Convergence. Calculus II (part 3): Sequences and Series (by Evan Dummit, 2015, v. 2. Calculus II (part 3): Sequeces ad Series (by Eva Dummit, 05, v..00) Cotets 7 Sequeces ad Series 7. Sequeces ad Covergece......................................... 7. Iite Series.................................................

More information

MATH 361 Homework 9. Royden Royden Royden

MATH 361 Homework 9. Royden Royden Royden MATH 61 Homework 9 Royde..9 First, we show that for ay subset E of the real umbers, E c + y = E + y) c traslatig the complemet is equivalet to the complemet of the traslated set). Without loss of geerality,

More information

A Simplified Binet Formula for k-generalized Fibonacci Numbers

A Simplified Binet Formula for k-generalized Fibonacci Numbers A Simplified Biet Formula for k-geeralized Fiboacci Numbers Gregory P. B. Dresde Departmet of Mathematics Washigto ad Lee Uiversity Lexigto, VA 440 dresdeg@wlu.edu Abstract I this paper, we preset a particularly

More information

MA2108S Tutorial 5 Solution

MA2108S Tutorial 5 Solution MA08S Tutorial 5 Solutio Prepared by: LuJigyi LuoYusheg March 0 Sectio 3. Questio 7. Let x := / l( + ) for N. (a). Use the difiitio of limit to show that lim(x ) = 0. Proof. Give ay ɛ > 0, sice ɛ > 0,

More information

SUMS OF n-th POWERS OF ROOTS OF A GIVEN QUADRATIC EQUATION. N.A. Draim, Ventura, Calif., and Marjorie Bicknell Wilcox High School, Santa Clara, Calif.

SUMS OF n-th POWERS OF ROOTS OF A GIVEN QUADRATIC EQUATION. N.A. Draim, Ventura, Calif., and Marjorie Bicknell Wilcox High School, Santa Clara, Calif. SUMS OF -th OWERS OF ROOTS OF A GIVEN QUADRATIC EQUATION N.A. Draim, Vetura, Calif., ad Marjorie Bickell Wilcox High School, Sata Clara, Calif. The quadratic equatio whose roots a r e the sum or differece

More information

3. Continuous Random Variables

3. Continuous Random Variables Statistics ad probability: 3-1 3. Cotiuous Radom Variables A cotiuous radom variable is a radom variable which ca take values measured o a cotiuous scale e.g. weights, stregths, times or legths. For ay

More information

The Stable Marriage Problem

The Stable Marriage Problem The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,

More information

A Gentle Introduction to Algorithms: Part II

A Gentle Introduction to Algorithms: Part II A Getle Itroductio to Algorithms: Part II Cotets of Part I:. Merge: (to merge two sorted lists ito a sigle sorted list.) 2. Bubble Sort 3. Merge Sort: 4. The Big-O, Big-Θ, Big-Ω otatios: asymptotic bouds

More information

Chapter 5: Inner Product Spaces

Chapter 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 information

The Poisson Distribution

The Poisson Distribution Lecture 5 The Poisso Distributio 5.1 Itroductio Example 5.1: Drowigs i Malta The book [Mou98] cites data from the St. Luke s Hospital Gazette, o the mothly umber of drowigs o Malta, over a period of early

More information

THE ABRACADABRA PROBLEM

THE 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 information

Stochastic Processes and Advanced Mathematical Finance. Laws of Large Numbers

Stochastic Processes and Advanced Mathematical Finance. Laws of Large Numbers Steven R. Dunbar Department of Mathematics 203 Avery Hall University of Nebraska-Lincoln Lincoln, NE 68588-0130 http://www.math.unl.edu Voice: 402-472-3731 Fax: 402-472-8466 Stochastic Processes and Advanced

More information

Solving Nonlinear Equation

Solving Nonlinear Equation Solvi Noliear Equatio oot r Noliear Equatios Give uctio, we id value or which Solutio is root o equatio, or zero o uctio So problem is kow as root idi or zero idi Numerical Methods We-Chieh Li Noliear

More information

Economics 140A Confidence Intervals and Hypothesis Testing

Economics 140A Confidence Intervals and Hypothesis Testing Ecoomics 140A Cofidece Itervals ad Hypothesis Testig Obtaiig a estimate of a parameter is ot the al purpose of statistical iferece because it is highly ulikely that the populatio value of a parameter is

More information

AQA STATISTICS 1 REVISION NOTES

AQA STATISTICS 1 REVISION NOTES AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if

More information

Hypergeometric Distributions

Hypergeometric Distributions 7.4 Hypergeometric Distributios Whe choosig the startig lie-up for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you

More information

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable

Week 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 information

Counting II 3, 7 3, 2 3, 9 7, 2 7, 9 2, 9

Counting II 3, 7 3, 2 3, 9 7, 2 7, 9 2, 9 Coutig II Sometimes we will wat to choose objects from a set of objects, ad we wo t be iterested i orderig them. For example, if you are leavig for vacatio ad you wat to pac your suitcase with three of

More information

THE COMPLETENESS OF CONVERGENT SEQUENCES SPACE OF FUZZY NUMBERS. Hee Chan Choi

THE COMPLETENESS OF CONVERGENT SEQUENCES SPACE OF FUZZY NUMBERS. Hee Chan Choi Kagweo-Kyugki Math. Jour. 4 (1996), No. 2, pp. 117 124 THE COMPLETENESS OF CONVERGENT SEQUENCES SPACE OF FUZZY NUMBERS Hee Cha Choi Abstract. I this paper we defie a ew fuzzy metric θ of fuzzy umber sequeces,

More information

Numerical Solution of Equations

Numerical Solution of Equations School of Mechaical Aerospace ad Civil Egieerig Numerical Solutio of Equatios T J Craft George Begg Buildig, C4 TPFE MSc CFD- Readig: J Ferziger, M Peric, Computatioal Methods for Fluid Dyamics HK Versteeg,

More information

Chapter 7 Methods of Finding Estimators

Chapter 7 Methods of Finding Estimators Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

More information

A Mathematical Perspective on Gambling

A Mathematical Perspective on Gambling A Mathematical Perspective o Gamblig Molly Maxwell Abstract. This paper presets some basic topics i probability ad statistics, icludig sample spaces, probabilistic evets, expectatios, the biomial ad ormal

More information

MEI Structured Mathematics. Module Summary Sheets. Statistics 2 (Version B: reference to new book)

MEI Structured Mathematics. Module Summary Sheets. Statistics 2 (Version B: reference to new book) MEI Mathematics i Educatio ad Idustry MEI Structured Mathematics Module Summary Sheets Statistics (Versio B: referece to ew book) Topic : The Poisso Distributio Topic : The Normal Distributio Topic 3:

More information

Central Limit Theorem and Its Applications to Baseball

Central Limit Theorem and Its Applications to Baseball Cetral Limit Theorem ad Its Applicatios to Baseball by Nicole Aderso A project submitted to the Departmet of Mathematical Scieces i coformity with the requiremets for Math 4301 (Hoours Semiar) Lakehead

More information

Advanced Probability Theory

Advanced Probability Theory Advaced Probability Theory Math5411 HKUST Kai Che (Istructor) Chapter 1. Law of Large Numbers 1.1. σ-algebra, measure, probability space ad radom variables. This sectio lays the ecessary rigorous foudatio

More information

7 b) 0. Guided Notes for lesson P.2 Properties of Exponents. If a, b, x, y and a, b, 0, and m, n Z then the following properties hold: 1 n b

7 b) 0. Guided Notes for lesson P.2 Properties of Exponents. If a, b, x, y and a, b, 0, and m, n Z then the following properties hold: 1 n b Guided Notes for lesso P. Properties of Expoets If a, b, x, y ad a, b, 0, ad m, Z the the followig properties hold:. Negative Expoet Rule: b ad b b b Aswers must ever cotai egative expoets. Examples: 5

More information

A Study for the (μ,s) n Relation for Tent Map

A Study for the (μ,s) n Relation for Tent Map Applied Mathematical Scieces, Vol. 8, 04, o. 60, 3009-305 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/ams.04.4437 A Study for the (μ,s) Relatio for Tet Map Saba Noori Majeed Departmet of Mathematics

More information

Measure Theory, MA 359 Handout 1

Measure Theory, MA 359 Handout 1 Measure Theory, M 359 Hadout 1 Valeriy Slastikov utum, 2005 1 Measure theory 1.1 Geeral costructio of Lebesgue measure I this sectio we will do the geeral costructio of σ-additive complete measure by extedig

More information