The Random Walk For Dummies
|
|
- Alaina Terry
- 6 years ago
- Views:
Transcription
1 The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli process ad the Catala umbers i greater depth Fially we determie the probability that, if a druk is foud agai at the bar, the this is his first retur visit Itroductio The radom walk has bee a topic of iterest i may disciplies, but it has bee of particular iterest i probability theory Ideed, every studet of probability theory has heard of the radom walk, especially i the form of a druk leavig a bar ad waderig aimlessly up ad dow the street Amog other issues, the followig four have bee ivestigated i oe, two, ad three dimesios: The expected positio x of the druk after steps The maximum positio x that the druk has reached after steps The expected time of the druk s last visit to The probability that the druk has t stumbled upo his ow path after steps These issues are discussed, for example, i Rota s book o probability [3] The oe-dimesioal discrete case is most widely kow because it is simple, yet illustrates may iterestig ad importat features I this paper, we treat this case of the first issue above Thus we aswer a basic questio: What is the probability that the druk is at a certai distace x, from the bar, after steps? I Sectio, we review five basic topics: the sample space, the biomial coefficiet, radom variables, the Beroulli process, ad Catala umbers I Sectio 3, we fid the probability distributio for the positio of the druk after steps I Sectio, we proceed to calculate the probability that the druk arrives at the bar for the first time after steps Fially, i Sectio 5, we determie the coditioal probability that the druk s first retur occurs o the th step give that he is ideed at the bar the; the formula is surprisigly simple The Basics I this sectio, we review five cocepts of probability theory, which we will use to study the radom walk of the druk First, a sample space Ω is defied to be the set of all possible outcomes of a experimet Cosider the coi toss as a example A sigle toss has two possible outcomes: heads H, or tails T ; thus Ω = {H, T } If we toss the coi twice, the there are differet possibilities; ow Ω = {T T, T H, HT, HH} Secod, the biomial coefficiet ( k) is defied to be the umber of k-combiatios of a -elemet set I other words, it is the umber of differet ways to pick k elemets out of The biomial coefficiet is give by the formula, ( )! = k ( k)! k! 3
2 MIT Udergraduate Joural of Mathematics This formula is proved i [, p 6] Third, a radom variable is defied to be a fuctio X that assigs to each elemet c, i the sample space of a experimet, oe ad oly oe real umber X(c) The sample space Ω of X is the set of real umbers x such that x = X(c) for some c i the sample space of the experimet This defiitio is foud i [, p ] Fourth, a Beroulli process is defied to be a sequece of radom variables X, X, X 3, Each X i records the outcome of a experimet modeled by the toss of a coi Let p equal the probability of gettig a head, ad q the probability of gettig a tail Sice these are the oly possible outcomes, p + q = Let X i be if the th trial yields a head, ad be if a tail Thus, the sample space Ω of a Beroulli process is the set of all possible sequeces of ad Fifth ad fially, the th Catala umber c couts certai arragemets of paretheses A well-formed arragemet is a list of paretheses where each ope parethesis ca be paired with a correspodig closed parethesis to its right The th Catala umber is the umber of such well-formed arragemets For example, whe = 3, the possible well-formed arragemets are ()()(), ()(()), (())(), (()()), ((())) Accordig to [, p 53], the th Catala umber c is give by the formula, c = ( ) + Thus, for example, the first six Catala umbers are,,, 5,, ad 3 How It All Ties I Suppose a druk leaves a bar ad walks aimlessly up ad dow the street, totally disorieted We model the street as a lie with the bar at the origi, ad assume that the druk takes uit steps, so we may record his positio with a iteger Thus, for example, if he takes 5 steps to the left, he will be at positio 5 We ow calculate the probability that the druk is at positio x after steps Assume that the druk s walk ca be modeled by a Beroulli process Say that the ith step is represeted by the radom variable X i A value of idicates a step to the left; a value of, a step to the right, so that G = X + X + + X gives the positio of the druk after steps We wat to kow the distributio of the radom variable G Deote the value of G by x Deote the umber of steps take to the right by r, the umber to the left by l The It follows that x = r l ad = r + l r = (x + ) ad l = ( x) Now, there are ( l) ways that l give steps ca occur amog total steps This is also the umber of ways of arrivig at the poit x, ad each way has probability p r q l Note that ad x must have the same parity because x = l We ca therefore coclude that the probability distributio at poit x is give by the formula,
3 The Radom Walk For Dummies 5 { ( ) P (G = x) = l p r q l, if x = mod ;, otherwise Cosider for example the case p = q = ; this is the case of the symmetric radom walk The probability P (G = x) of beig at positio x after steps is give i Table 3- This table is simply a pascal triagle iterspersed with s Table 3- The Symmetric Radom Walk \ x The probability distributio i this particular case is give by ( )( ) ( )/ P (G = x) = = l l Sice every path is equally likely i the symmetric radom walk, the probability ca be iterpreted purely combiatorially The probability is the umber of differet ways of arrivig at x divided by the size of the sample space The sample space is the set of all the possible paths of legth Sice the druk has two choices at each poit, ad he takes a total of steps, the total umber of possibilities is Takig a Step Further From ow o, we assume that the radom walk is symmetric; that is, p = What is the probability that the druk s first retur is at the th step? Here is where Catala umbers eter Observe that there is a oe-to-oe correspodece betwee paths ad arragemets of paretheses First, let a ope parethesis represet a step to the left, ad a closed parethesis a step to the right For ow, we cosider oly the case where the druk s first step is towards the left, sice the case to the right is clearly symmetric to it From Sectio, recall the defiitio of the Catala umber c : it is the umber of well-formed arragemets of paretheses Note that, for well-formed arragemets, the umber of ope paretheses is always at least that of closed paretheses, regardless of what first k parethesis we pick Because of the correspodece betwee paths ad arragemets, the Catala umbers cout the umber of paths that the druk ca take, which start ad ed at the bar, without ever crossig to the right side of the bar We ow adapt our correspodece to the problem of the druk s first retur The druk takes his first step to the left O the ext steps, we isist that the druk s path correspods to a well-formed arragemet so that at step he is at positio
4 6 MIT Udergraduate Joural of Mathematics without ever crossig to the right of The the fial step brigs the druk back ito the bar This couts the total umber of differet paths that the druk ca take give the coditio that the first retur be at the th step Summarizig, we have the followig sequece of steps: Step The druk takes oe step to the left The ext - steps The druk follows a path, which correspods to a well-formed arragemet, that restricts the druk to be either at poit or to the left of it, ad that forces him to be at after the ( )st step Step The druk s first retur visit occurs whe he takes oe step to the right ad ito the bar Hece, the umber of paths the druk ca take is simply the Catala umber c, sice the well-ordered restrictio applies oly to the middle steps We kow that the th Catala umber is equal to + ( ) So the ( )st Catala umber is simply ( ) Allowig the druk to take his first step to the right will double the total umber of paths that the druk ca take So the total is ( ) To compute the probability of first retur, we eed oly divide this quatity by the size of the total sample space The latter was see at the ed of Sectio 3 to be sice there are steps Therefore, the probability that the druk reaches the bar for the first time after steps is ( ) / So, for example, cosider the probability that the first retur is at Step = 6; this probablity is give by ( )/ 6 = Coditioal Probability ad the First Retur We ow cosider a variat of the problem studied i the previous sectio: Give that the druk is at the bar at Step, what is the probability that this is his first retur visit? This problem is like the previous oe The oly differece is that the sample space has bee reduced Istead of cosiderig all of the possible paths, we ow cosider the total umber of paths give that, i the ed, the druk will be at the bar, which he may or may ot have passed by earlier
5 The Radom Walk For Dummies 7 To fid the probability that the druk will be at the bar after steps, we use the coclusios from Sectio 3 to obtai ( ) P (G = ) = p () q () Sice p = q =, this expressio becomes ( ) / The umerator i this expressio represets the umber of paths the druk ca take provided he is at the bar at the th step This is sample space we eed for the coditioal probability From Sectio, recall that, if the druk s first retur visit is o the th step, the the umber of paths that the druk ca take is ( ) O the other had, i Sectio 3, we foud the umber of paths that the druk ca take that put him back at the bar o Step ; this umber is simply ( ) Fially we divide these two umbers, gettig ( ) / ( ) = / ( )! ()! ( )!( )!!!, which simplifies to / ( ) = ( ) = The fial simplicity is amazig! The formula is just ad is uexpectedly simple! #(steps), Refereces [] Brualdi, R, Itroductory Combiatorics, Pretice-Hall, Third Editio, 999 [] Hogg, Robert V, Itroductio to Mathematical Statistics, Fifth Editio, 995 [3] Rota, G C, Probability Theory, Prelimiary Editio, 99
6 This page will be blak
Random Models. Tusheng Zhang. February 14, 2013
Radom Models Tusheg Zhag February 14, 013 1 Radom Walks Let me describe the model. Radom walks are used to describe the motio of a movig particle (object). Suppose that a particle (object) moves alog the
More informationInjections, Surjections, and the Pigeonhole Principle
Ijectios, Surjectios, ad the Pigeohole Priciple 1 (10 poits Here we will come up with a sloppy boud o the umber of parethesisestigs (a (5 poits Describe a ijectio from the set of possible ways to est pairs
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More informationLecture 4 The Simple Random Walk
Lecture 4: The Simple Radom Walk 1 of 9 Course: M36K Itro to Stochastic Processes Term: Fall 014 Istructor: Gorda Zitkovic Lecture 4 The Simple Radom Walk We have defied ad costructed a radom walk {X }
More informationLecture 2: April 3, 2013
TTIC/CMSC 350 Mathematical Toolkit Sprig 203 Madhur Tulsiai Lecture 2: April 3, 203 Scribe: Shubhedu Trivedi Coi tosses cotiued We retur to the coi tossig example from the last lecture agai: Example. Give,
More informationPH 425 Quantum Measurement and Spin Winter SPINS Lab 1
PH 425 Quatum Measuremet ad Spi Witer 23 SPIS Lab Measure the spi projectio S z alog the z-axis This is the experimet that is ready to go whe you start the program, as show below Each atom is measured
More informationProblems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman:
Math 224 Fall 2017 Homework 4 Drew Armstrog Problems from 9th editio of Probability ad Statistical Iferece by Hogg, Tais ad Zimmerma: Sectio 2.3, Exercises 16(a,d),18. Sectio 2.4, Exercises 13, 14. Sectio
More informationLecture Overview. 2 Permutations and Combinations. n(n 1) (n (k 1)) = n(n 1) (n k + 1) =
COMPSCI 230: Discrete Mathematics for Computer Sciece April 8, 2019 Lecturer: Debmalya Paigrahi Lecture 22 Scribe: Kevi Su 1 Overview I this lecture, we begi studyig the fudametals of coutig discrete objects.
More informationAdvanced Stochastic Processes.
Advaced Stochastic Processes. David Gamarik LECTURE 2 Radom variables ad measurable fuctios. Strog Law of Large Numbers (SLLN). Scary stuff cotiued... Outlie of Lecture Radom variables ad measurable fuctios.
More informationSection 5.1 The Basics of Counting
1 Sectio 5.1 The Basics of Coutig Combiatorics, the study of arragemets of objects, is a importat part of discrete mathematics. I this chapter, we will lear basic techiques of coutig which has a lot of
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
More informationStatistics 511 Additional Materials
Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability
More informationAnalytic Theory of Probabilities
Aalytic Theory of Probabilities PS Laplace Book II Chapter II, 4 pp 94 03 4 A lottery beig composed of umbered tickets of which r exit at each drawig, oe requires the probability that after i drawigs all
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 018/019 DR. ANTHONY BROWN 8. Statistics 8.1. Measures of Cetre: Mea, Media ad Mode. If we have a series of umbers the
More informationDiscrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 22
CS 70 Discrete Mathematics for CS Sprig 2007 Luca Trevisa Lecture 22 Aother Importat Distributio The Geometric Distributio Questio: A biased coi with Heads probability p is tossed repeatedly util the first
More informationCALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
More informationGoodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen)
Goodess-of-Fit Tests ad Categorical Data Aalysis (Devore Chapter Fourtee) MATH-252-01: Probability ad Statistics II Sprig 2019 Cotets 1 Chi-Squared Tests with Kow Probabilities 1 1.1 Chi-Squared Testig................
More informationPUTNAM TRAINING PROBABILITY
PUTNAM TRAINING PROBABILITY (Last udated: December, 207) Remark. This is a list of exercises o robability. Miguel A. Lerma Exercises. Prove that the umber of subsets of {, 2,..., } with odd cardiality
More informationIt is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function.
MATH 532 Measurable Fuctios Dr. Neal, WKU Throughout, let ( X, F, µ) be a measure space ad let (!, F, P ) deote the special case of a probability space. We shall ow begi to study real-valued fuctios defied
More informationsubcaptionfont+=small,labelformat=parens,labelsep=space,skip=6pt,list=0,hypcap=0 subcaption ALGEBRAIC COMBINATORICS LECTURE 8 TUESDAY, 2/16/2016
subcaptiofot+=small,labelformat=pares,labelsep=space,skip=6pt,list=0,hypcap=0 subcaptio ALGEBRAIC COMBINATORICS LECTURE 8 TUESDAY, /6/06. Self-cojugate Partitios Recall that, give a partitio λ, we may
More informationSection 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations
Differece Equatios to Differetial Equatios Sectio. Calculus: Areas Ad Tagets The study of calculus begis with questios about chage. What happes to the velocity of a swigig pedulum as its positio chages?
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationFinal Review for MATH 3510
Fial Review for MATH 50 Calculatio 5 Give a fairly simple probability mass fuctio or probability desity fuctio of a radom variable, you should be able to compute the expected value ad variace of the variable
More informationIE 230 Seat # Name < KEY > Please read these directions. Closed book and notes. 60 minutes.
IE 230 Seat # Name < KEY > Please read these directios. Closed book ad otes. 60 miutes. Covers through the ormal distributio, Sectio 4.7 of Motgomery ad Ruger, fourth editio. Cover page ad four pages of
More information(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?
MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle
More informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals
More informationChapter 1 : Combinatorial Analysis
STAT/MATH 394 A - PROBABILITY I UW Autum Quarter 205 Néhémy Lim Chapter : Combiatorial Aalysis A major brach of combiatorial aalysis called eumerative combiatorics cosists of studyig methods for coutig
More informationCIS Spring 2018 (instructor Val Tannen)
CIS 160 - Sprig 2018 (istructor Val Tae) Lecture 5 Thursday, Jauary 25 COUNTING We cotiue studyig how to use combiatios ad what are their properties. Example 5.1 How may 8-letter strigs ca be costructed
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit theorems Throughout this sectio we will assume a probability space (Ω, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationBooks Recommended for Further Reading
Books Recommeded for Further Readig by 8.5..8 o 0//8. For persoal use oly.. K. P. Bogart, Itroductory Combiatorics rd ed., S. I. Harcourt Brace College Publishers, 998.. R. A. Brualdi, Itroductory Combiatorics
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationECEN 644 HOMEWORK #5 SOLUTION SET
ECE 644 HOMEWORK #5 SOUTIO SET 7. x is a real valued sequece. The first five poits of its 8-poit DFT are: {0.5, 0.5 - j 0.308, 0, 0.5 - j 0.058, 0} To compute the 3 remaiig poits, we ca use the followig
More informationCounting Well-Formed Parenthesizations Easily
Coutig Well-Formed Parethesizatios Easily Pekka Kilpeläie Uiversity of Easter Filad School of Computig, Kuopio August 20, 2014 Abstract It is well kow that there is a oe-to-oe correspodece betwee ordered
More informationn outcome is (+1,+1, 1,..., 1). Let the r.v. X denote our position (relative to our starting point 0) after n moves. Thus X = X 1 + X 2 + +X n,
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 9 Variace Questio: At each time step, I flip a fair coi. If it comes up Heads, I walk oe step to the right; if it comes up Tails, I walk oe
More informationUNIT 2 DIFFERENT APPROACHES TO PROBABILITY THEORY
UNIT 2 DIFFERENT APPROACHES TO PROBABILITY THEORY Structure 2.1 Itroductio Objectives 2.2 Relative Frequecy Approach ad Statistical Probability 2. Problems Based o Relative Frequecy 2.4 Subjective Approach
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationBertrand s Postulate
Bertrad s Postulate Lola Thompso Ross Program July 3, 2009 Lola Thompso (Ross Program Bertrad s Postulate July 3, 2009 1 / 33 Bertrad s Postulate I ve said it oce ad I ll say it agai: There s always a
More informationGG313 GEOLOGICAL DATA ANALYSIS
GG313 GEOLOGICAL DATA ANALYSIS 1 Testig Hypothesis GG313 GEOLOGICAL DATA ANALYSIS LECTURE NOTES PAUL WESSEL SECTION TESTING OF HYPOTHESES Much of statistics is cocered with testig hypothesis agaist data
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationThe Binomial Theorem
The Biomial Theorem Robert Marti Itroductio The Biomial Theorem is used to expad biomials, that is, brackets cosistig of two distict terms The formula for the Biomial Theorem is as follows: (a + b ( k
More information4. Partial Sums and the Central Limit Theorem
1 of 10 7/16/2009 6:05 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 4. Partial Sums ad the Cetral Limit Theorem The cetral limit theorem ad the law of large umbers are the two fudametal theorems
More information18th Bay Area Mathematical Olympiad. Problems and Solutions. February 23, 2016
18th Bay Area Mathematical Olympiad February 3, 016 Problems ad Solutios BAMO-8 ad BAMO-1 are each 5-questio essay-proof exams, for middle- ad high-school studets, respectively. The problems i each exam
More informationJanuary 25, 2017 INTRODUCTION TO MATHEMATICAL STATISTICS
Jauary 25, 207 INTRODUCTION TO MATHEMATICAL STATISTICS Abstract. A basic itroductio to statistics assumig kowledge of probability theory.. Probability I a typical udergraduate problem i probability, we
More informationProbability and statistics: basic terms
Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample
More informationFrequentist Inference
Frequetist Iferece The topics of the ext three sectios are useful applicatios of the Cetral Limit Theorem. Without kowig aythig about the uderlyig distributio of a sequece of radom variables {X i }, for
More informationAnalysis of Algorithms. Introduction. Contents
Itroductio The focus of this module is mathematical aspects of algorithms. Our mai focus is aalysis of algorithms, which meas evaluatig efficiecy of algorithms by aalytical ad mathematical methods. We
More informationSEQUENCES AND SERIES
Sequeces ad 6 Sequeces Ad SEQUENCES AND SERIES Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives rise to what is called a sequece. Sequeces
More informationProperties and Tests of Zeros of Polynomial Functions
Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology
More informationLet us consider the following problem to warm up towards a more general statement.
Lecture 4: Sequeces with repetitios, distributig idetical objects amog distict parties, the biomial theorem, ad some properties of biomial coefficiets Refereces: Relevat parts of chapter 15 of the Math
More informationLecture 12: November 13, 2018
Mathematical Toolkit Autum 2018 Lecturer: Madhur Tulsiai Lecture 12: November 13, 2018 1 Radomized polyomial idetity testig We will use our kowledge of coditioal probability to prove the followig lemma,
More informationMath 105 TOPICS IN MATHEMATICS REVIEW OF LECTURES VII. 7. Binomial formula. Three lectures ago ( in Review of Lectuires IV ), we have covered
Math 5 TOPICS IN MATHEMATICS REVIEW OF LECTURES VII Istructor: Lie #: 59 Yasuyuki Kachi 7 Biomial formula February 4 Wed) 5 Three lectures ago i Review of Lectuires IV ) we have covered / \ / \ / \ / \
More informationCSE 191, Class Note 05: Counting Methods Computer Sci & Eng Dept SUNY Buffalo
Coutig Methods CSE 191, Class Note 05: Coutig Methods Computer Sci & Eg Dept SUNY Buffalo c Xi He (Uiversity at Buffalo CSE 191 Discrete Structures 1 / 48 Need for Coutig The problem of coutig the umber
More informationAxioms of Measure Theory
MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so that
More informationDiscrete probability distributions
Discrete probability distributios I the chapter o probability we used the classical method to calculate the probability of various values of a radom variable. I some cases, however, we may be able to develop
More informationDiscrete Mathematics for CS Spring 2005 Clancy/Wagner Notes 21. Some Important Distributions
CS 70 Discrete Mathematics for CS Sprig 2005 Clacy/Wager Notes 21 Some Importat Distributios Questio: A biased coi with Heads probability p is tossed repeatedly util the first Head appears. What is the
More informationPhysics 116A Solutions to Homework Set #1 Winter Boas, problem Use equation 1.8 to find a fraction describing
Physics 6A Solutios to Homework Set # Witer 0. Boas, problem. 8 Use equatio.8 to fid a fractio describig 0.694444444... Start with the formula S = a, ad otice that we ca remove ay umber of r fiite decimals
More informationMeasures of Spread: Standard Deviation
Measures of Spread: Stadard Deviatio So far i our study of umerical measures used to describe data sets, we have focused o the mea ad the media. These measures of ceter tell us the most typical value of
More informationProduct measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.
Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the
More information5. Likelihood Ratio Tests
1 of 5 7/29/2009 3:16 PM Virtual Laboratories > 9. Hy pothesis Testig > 1 2 3 4 5 6 7 5. Likelihood Ratio Tests Prelimiaries As usual, our startig poit is a radom experimet with a uderlyig sample space,
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationEstimation for Complete Data
Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of
More informationx a x a Lecture 2 Series (See Chapter 1 in Boas)
Lecture Series (See Chapter i Boas) A basic ad very powerful (if pedestria, recall we are lazy AD smart) way to solve ay differetial (or itegral) equatio is via a series expasio of the correspodig solutio
More informationSection 11.8: Power Series
Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i
More informationTopic 5: Basics of Probability
Topic 5: Jue 1, 2011 1 Itroductio Mathematical structures lie Euclidea geometry or algebraic fields are defied by a set of axioms. Mathematical reality is the developed through the itroductio of cocepts
More information1 Generating functions for balls in boxes
Math 566 Fall 05 Some otes o geeratig fuctios Give a sequece a 0, a, a,..., a,..., a geeratig fuctio some way of represetig the sequece as a fuctio. There are may ways to do this, with the most commo ways
More informationProbability theory and mathematical statistics:
N.I. Lobachevsky State Uiversity of Nizhi Novgorod Probability theory ad mathematical statistics: Law of Total Probability. Associate Professor A.V. Zorie Law of Total Probability. 1 / 14 Theorem Let H
More information6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer.
6 Itegers Modulo I Example 2.3(e), we have defied the cogruece of two itegers a,b with respect to a modulus. Let us recall that a b (mod ) meas a b. We have proved that cogruece is a equivalece relatio
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 19
CS 70 Discrete Mathematics ad Probability Theory Sprig 2016 Rao ad Walrad Note 19 Some Importat Distributios Recall our basic probabilistic experimet of tossig a biased coi times. This is a very simple
More informationWorksheet on Generating Functions
Worksheet o Geeratig Fuctios October 26, 205 This worksheet is adapted from otes/exercises by Nat Thiem. Derivatives of Geeratig Fuctios. If the sequece a 0, a, a 2,... has ordiary geeratig fuctio A(x,
More informationChapter 7 COMBINATIONS AND PERMUTATIONS. where we have the specific formula for the binomial coefficients:
Chapter 7 COMBINATIONS AND PERMUTATIONS We have see i the previous chapter that (a + b) ca be writte as 0 a % a & b%þ% a & b %þ% b where we have the specific formula for the biomial coefficiets: '!!(&)!
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationThe multiplicative structure of finite field and a construction of LRC
IERG6120 Codig for Distributed Storage Systems Lecture 8-06/10/2016 The multiplicative structure of fiite field ad a costructio of LRC Lecturer: Keeth Shum Scribe: Zhouyi Hu Notatios: We use the otatio
More informationOn Random Line Segments in the Unit Square
O Radom Lie Segmets i the Uit Square Thomas A. Courtade Departmet of Electrical Egieerig Uiversity of Califoria Los Ageles, Califoria 90095 Email: tacourta@ee.ucla.edu I. INTRODUCTION Let Q = [0, 1] [0,
More information1 Approximating Integrals using Taylor Polynomials
Seughee Ye Ma 8: Week 7 Nov Week 7 Summary This week, we will lear how we ca approximate itegrals usig Taylor series ad umerical methods. Topics Page Approximatig Itegrals usig Taylor Polyomials. Defiitios................................................
More informationAnalysis of Experimental Measurements
Aalysis of Experimetal Measuremets Thik carefully about the process of makig a measuremet. A measuremet is a compariso betwee some ukow physical quatity ad a stadard of that physical quatity. As a example,
More informationIntermediate Math Circles November 4, 2009 Counting II
Uiversity of Waterloo Faculty of Mathematics Cetre for Educatio i Mathematics ad Computig Itermediate Math Circles November 4, 009 Coutig II Last time, after lookig at the product rule ad sum rule, we
More informationOptimally Sparse SVMs
A. Proof of Lemma 3. We here prove a lower boud o the umber of support vectors to achieve geeralizatio bouds of the form which we cosider. Importatly, this result holds ot oly for liear classifiers, but
More informationLecture 3: Catalan Numbers
CCS Discrete Math I Professor: Padraic Bartlett Lecture 3: Catala Numbers Week 3 UCSB 2014 I this week, we start studyig specific examples of commoly-occurrig sequeces of umbers (as opposed to the more
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you
More informationMachine Learning for Data Science (CS 4786)
Machie Learig for Data Sciece CS 4786) Lecture & 3: Pricipal Compoet Aalysis The text i black outlies high level ideas. The text i blue provides simple mathematical details to derive or get to the algorithm
More informationCombinatorics and Newton s theorem
INTRODUCTION TO MATHEMATICAL REASONING Key Ideas Worksheet 5 Combiatorics ad Newto s theorem This week we are goig to explore Newto s biomial expasio theorem. This is a very useful tool i aalysis, but
More informationINFINITE SEQUENCES AND SERIES
11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS
More informationEnd-of-Year Contest. ERHS Math Club. May 5, 2009
Ed-of-Year Cotest ERHS Math Club May 5, 009 Problem 1: There are 9 cois. Oe is fake ad weighs a little less tha the others. Fid the fake coi by weighigs. Solutio: Separate the 9 cois ito 3 groups (A, B,
More informationMath 61CM - Solutions to homework 3
Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig
More informationRandomized Algorithms I, Spring 2018, Department of Computer Science, University of Helsinki Homework 1: Solutions (Discussed January 25, 2018)
Radomized Algorithms I, Sprig 08, Departmet of Computer Sciece, Uiversity of Helsiki Homework : Solutios Discussed Jauary 5, 08). Exercise.: Cosider the followig balls-ad-bi game. We start with oe black
More informationPutnam Training Exercise Counting, Probability, Pigeonhole Principle (Answers)
Putam Traiig Exercise Coutig, Probability, Pigeohole Pricile (Aswers) November 24th, 2015 1. Fid the umber of iteger o-egative solutios to the followig Diohatie equatio: x 1 + x 2 + x 3 + x 4 + x 5 = 17.
More informationApproximations and more PMFs and PDFs
Approximatios ad more PMFs ad PDFs Saad Meimeh 1 Approximatio of biomial with Poisso Cosider the biomial distributio ( b(k,,p = p k (1 p k, k λ: k Assume that is large, ad p is small, but p λ at the limit.
More informationMa 530 Introduction to Power Series
Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power
More informationDiscrete-Time Systems, LTI Systems, and Discrete-Time Convolution
EEL5: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we begi our mathematical treatmet of discrete-time s. As show i Figure, a discrete-time operates or trasforms some iput sequece x [
More informationCommutativity in Permutation Groups
Commutativity i Permutatio Groups Richard Wito, PhD Abstract I the group Sym(S) of permutatios o a oempty set S, fixed poits ad trasiet poits are defied Prelimiary results o fixed ad trasiet poits are
More informationSNAP Centre Workshop. Basic Algebraic Manipulation
SNAP Cetre Workshop Basic Algebraic Maipulatio 8 Simplifyig Algebraic Expressios Whe a expressio is writte i the most compact maer possible, it is cosidered to be simplified. Not Simplified: x(x + 4x)
More informationThe Binomial Theorem
The Biomial Theorem Lecture 47 Sectio 9.7 Robb T. Koether Hampde-Sydey College Fri, Apr 8, 204 Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, 204 / 25 Combiatios 2 Pascal s Triagle
More informationPlease do NOT write in this box. Multiple Choice. Total
Istructor: Math 0560, Worksheet Alteratig Series Jauary, 3000 For realistic exam practice solve these problems without lookig at your book ad without usig a calculator. Multiple choice questios should
More informationThe Riemann Zeta Function
Physics 6A Witer 6 The Riema Zeta Fuctio I this ote, I will sketch some of the mai properties of the Riema zeta fuctio, ζ(x). For x >, we defie ζ(x) =, x >. () x = For x, this sum diverges. However, we
More information