The Binomial Theorem
|
|
|
- Edwin Willis Banks
- 5 years ago
- Views:
Transcription
1 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
2 Combiatios 2 Pascal s Triagle 3 The Biomial Theorem 4 Biomial Probabilities 5 Assigmet Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
3 Outlie Combiatios 2 Pascal s Triagle 3 The Biomial Theorem 4 Biomial Probabilities 5 Assigmet Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
4 Combiatios Theorem Let ad r be oegative itegers with r. The ( ( =. r r Proof. To choose which r elemets to iclude i the subset is the same as choosig which r elemets ot to iclude. Thus, ( ( r = r. Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
5 A Recurrece Relatio Theorem Let ad r be positive itegers with r <. The ( ( ( = +. r r r Proof. Let A be a set of elemets ad let x A. Divide the subsets of size r ito two groups: ( Those that cotai x. (2 Those that do ot cotai x. Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
6 A Recurrece Relatio Proof. How may subsets are i group (? Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
7 A Recurrece Relatio Proof. How may subsets are i group (? If we remove x from each, we have all possible subsets of r elemets from A {x}, a set of elemets. So, there are ( r such subsets. Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
8 A Recurrece Relatio Proof. How may subsets are i group (? If we remove x from each, we have all possible subsets of r elemets from A {x}, a set of elemets. So, there are ( r such subsets. How may subsets are i group (2? Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
9 A Recurrece Relatio Proof. How may subsets are i group (? If we remove x from each, we have all possible subsets of r elemets from A {x}, a set of elemets. So, there are ( r such subsets. How may subsets are i group (2? The elemet x is i oe of them, so if we remove x from A, these subsets are all possible subsets of r elemets from A {x}, a set of elemets. So, there are ( r such subsets. Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
10 A Recurrece Relatio Proof. How may subsets are i group (? If we remove x from each, we have all possible subsets of r elemets from A {x}, a set of elemets. So, there are ( r such subsets. How may subsets are i group (2? The elemet x is i oe of them, so if we remove x from A, these subsets are all possible subsets of r elemets from A {x}, a set of elemets. So, there are ( such subsets. Thus, ( r ( = ( + r r r. Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
11 Outlie Combiatios 2 Pascal s Triagle 3 The Biomial Theorem 4 Biomial Probabilities 5 Assigmet Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
12 Pascal s Triagle The equatio ( = r ( r + ( r allows us to compute ( r recursively. The recursio eds with the boudary cases ( 0 = ad ( =. This is the basis of Pascal s Triagle. Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
13 Pascal s Triagle r Iitialize the boudary to Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
14 Pascal s Triagle r Compute ( 3 2 Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
15 Pascal s Triagle r Compute ( ( 4 2 ad 4 3 Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
16 Pascal s Triagle r Compute ( ( 5 2, 5 ( 3, ad 5 4 Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
17 Outlie Combiatios 2 Pascal s Triagle 3 The Biomial Theorem 4 Biomial Probabilities 5 Assigmet Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
18 The Biomial Theorem Theorem Let be a oegative iteger ad let a ad b be ay real umbers. The ( (a + b = a + a b + ( = a i b i. i i=0 ( ( a 2 b ab + b 2 Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, 204 / 25
19 The Biomial Theorem Proof. The proof is by iductio o. Whe = 0, we have (a + b 0 = ad 0 i=0 ( a i b i = i =. ( 0 a 0 0 b 0 0 Therefore, the statemet is true whe = 0. Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
20 The Biomial Theorem Proof. Suppose that the statemet is true for some iteger k where k 0. The (a + b = (a + b(a + b ( = (a + b a i b i i i=0 ( ( = a i b i + a i b i+ i i i=0 i=0 ( ( = a i b i + a i b i i i i=0 i= Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
21 The Biomial Theorem Proof. ( = a + a i b i + i i= i= [ ( = a + + i i= ( = a + a i b i + b i = i= ( a i b i. i i=0 ( a i b i + b i ( ] a i b i + b i Therefore, the statemet is true whe = k +, ad so it is true for all 0. Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
22 Examples Expad (a + b 5. Expad (a b 5. Expad (a + 2b 5. Show that ( ( 0 + ( + ( = 2. Show that ( ( 0 ( + ( 2 ± = 0. What is the value of ( ( ( (? What is the value of ( ( ( 2 ± 2 (? Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
23 Outlie Combiatios 2 Pascal s Triagle 3 The Biomial Theorem 4 Biomial Probabilities 5 Assigmet Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
24 Beroulli Trials Defiitio (Beroulli Trial A Beroulli trial is a experimet that has exactly two possible outcomes. The outcomes are called success ad failure. Toss a coi. Outcomes: heads or tails. Roll a die. Outcomes: eve or odd. Draw a card. Outcomes: ace or ot ace. Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
25 Biomial Experimets Defiitio (Biomial Experimet A biomial experimet is a experimet that cosists of a fixed umber of idepedet ad idetical Beroulli trials. Let be the umber of trials ad let p be the probability of success. Defiitio (Biomial Radom Variable A biomial radom variable is a variable whose value is the umber of successes i a biomial experimet. Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
26 Examples Toss a coi 0 times. Let X be the umber of heads. Roll a die 6 times. Let X be the umber of eve rolls. Draw 4 cards. Let X be the umber of aces. (Is this biomial? Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
27 Biomial Probability Distributio Proof. There are exactly ( r distict patters of successes ad failures i the trails. Each patter has the same probability, amely, p r ( p r. Therefore, the probability of oe of the patters occurrig is ( p r ( p r. r Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
28 Biomial Probability Distributio Theorem Let X be a biomial radom variable with parameters ad p. The the probability of exactly r successes is ( P(X = r = p r ( p r. r Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
29 Examples Toss a coi 0 times. What is the probability of exactly 6 heads? Roll a die 6 times. What is the probability of gettig a or a 2 exactly 3 times? Guess at all 25 aswers o a multiple-choice test with 5 choices for each aswer. What is the probability of scorig betwee 3 ad 7 correct? Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
30 Outlie Combiatios 2 Pascal s Triagle 3 The Biomial Theorem 4 Biomial Probabilities 5 Assigmet Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
31 Collected Collected Sec. 9.2: 7, 39bd, 42. Sec. 9.3: 2, 22. Sec. 9.4: 27. Sec. 9.5: 0, 2, 32. Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
32 Assigmet Assigmet Read Sectios 9.7, pages Exercises 0,, 2, 6, 8, 22, 26, 30, 32, 39, 42, page 603. Robb T. Koether (Hampde-Sydey College The Biomial Theorem Fri, Apr 8, / 25
The Binomial Theorem
The Biomial Theorem Lecture 47 Sectio 9.7 Robb T. Koether Hampde-Sydey College Thu, Apr 8, 03 Robb T. Koether (Hampde-Sydey College The Biomial Theorem Thu, Apr 8, 03 / 7 Combiatios Pascal s Triagle 3
AMS570 Lecture Notes #2
AMS570 Lecture Notes # Review of Probability (cotiued) Probability distributios. () Biomial distributio Biomial Experimet: ) It cosists of trials ) Each trial results i of possible outcomes, S or F 3)
Lecture 7: Properties of Random Samples
Lecture 7: Properties of Radom Samples 1 Cotiued From Last Class Theorem 1.1. Let X 1, X,...X be a radom sample from a populatio with mea µ ad variace σ
Assignment ( ) Class-XI. = iii. v. A B= A B '
Assigmet (8-9) Class-XI. Proe that: ( A B)' = A' B ' i A ( BAC) = ( A B) ( A C) ii A ( B C) = ( A B) ( A C) iv. A B= A B= φ v. A B= A B ' v A B B ' A'. A relatio R is dified o the set z of itegers as:
Some discrete distribution
Some discrete distributio p. 2-13 Defiitio (Beroulli distributio B(p)) A Beroulli distributio takes o oly two values: 0 ad 1, with probabilities 1 p ad p, respectively. pmf: p() = p (1 p) (1 ), if =0or
As stated by Laplace, Probability is common sense reduced to calculation.
Note: Hadouts DO NOT replace the book. I most cases, they oly provide a guidelie o topics ad a ituitive feel. The math details will be covered i class, so it is importat to atted class ad also you MUST
The Random Walk For Dummies
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
Stat 198 for 134 Instructor: Mike Leong
Chapter 2: Repeated Trials ad Samplig Sectio 2.1 Biomial Distributio 2.2 Normal Approximatio: Method 2.3 Normal Approximatios: Derivatio (Skip) 2.4 Poisso Approximatio 2.5 Radom Samplig Chapter 2 Table
f X (12) = Pr(X = 12) = Pr({(6, 6)}) = 1/36
Probability Distributios A Example With Dice If X is a radom variable o sample space S, the the probability that X takes o the value c is Similarly, Pr(X = c) = Pr({s S X(s) = c}) Pr(X c) = Pr({s S X(s)
Chapter 1. Probability
Chapter. Probability. Set defiitios 2. Set operatios 3. Probability itroduced through sets ad relative frequecy 4. Joit ad coditioal probability 5. Idepedet evets 6. Combied experimets 7. Beroulli trials
Combinatorics 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
Introduction to Probability and Statistics Twelfth Edition
Itroductio to Probability ad Statistics Twelfth Editio Robert J. Beaver Barbara M. Beaver William Medehall Presetatio desiged ad writte by: Barbara M. Beaver Itroductio to Probability ad Statistics Twelfth
PUTNAM 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
What is Probability?
Quatificatio of ucertaity. What is Probability? Mathematical model for thigs that occur radomly. Radom ot haphazard, do t kow what will happe o ay oe experimet, but has a log ru order. The cocept of probability
In number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.
Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers
M A T H F A L L CORRECTION. Algebra I 1 4 / 1 0 / U N I V E R S I T Y O F T O R O N T O
M A T H 2 4 0 F A L L 2 0 1 4 HOMEWORK ASSIGNMENT #4 CORRECTION Algebra I 1 4 / 1 0 / 2 0 1 4 U N I V E R S I T Y O F T O R O N T O P r o f e s s o r : D r o r B a r - N a t a Correctio Homework Assigmet
Math 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 {,,,
CSE 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
Homework 3. = k 1. Let S be a set of n elements, and let a, b, c be distinct elements of S. The number of k-subsets of S is
Homewor 3 Chapter 5 pp53: 3 40 45 Chapter 6 p85: 4 6 4 30 Use combiatorial reasoig to prove the idetity 3 3 Proof Let S be a set of elemets ad let a b c be distict elemets of S The umber of -subsets of
Sample Midterm This midterm consists of 10 questions. The rst seven questions are multiple choice; the remaining three
CS{74 Combiatorics & Discrete Probability, Fall 97 Sample Midterm :30{:00pm, 7 October Read these istructios carefully. This is a closed book exam. Calculators are permitted.. This midterm cosists of 0
What is Probability?
Quatificatio of ucertaity. What is Probability? Mathematical model for thigs that occur radomly. Radom ot haphazard, do t ow what will happe o ay oe experimet, but has a log ru order. The cocept of probability
Sequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
Final 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
f X (12) = Pr(X = 12) = Pr({(6, 6)}) = 1/36
Probability Distributios A Example With Dice If X is a radom variable o sample space S, the the probablity that X takes o the value c is Similarly, Pr(X = c) = Pr({s S X(s) = c} Pr(X c) = Pr({s S X(s)
1. n! = n. tion. For example, (n+1)! working with factorials. = (n+1) n (n 1) 2 1
Biomial Coefficiets ad Permutatios Mii-lecture The followig pages discuss a few special iteger coutig fuctios You may have see some of these before i a basic probability class or elsewhere, but perhaps
Intermediate 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
Fermat s Little Theorem. mod 13 = 0, = }{{} mod 13 = 0. = a a a }{{} mod 13 = a 12 mod 13 = 1, mod 13 = a 13 mod 13 = a.
Departmet of Mathematical Scieces Istructor: Daiva Puciskaite Discrete Mathematics Fermat s Little Theorem 43.. For all a Z 3, calculate a 2 ad a 3. Case a = 0. 0 0 2-times Case a 0. 0 0 3-times a a 2-times
CALCULATION 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
You may work in pairs or purely individually for this assignment.
CS 04 Problem Solvig i Computer Sciece OOC Assigmet 6: Recurreces You may work i pairs or purely idividually for this assigmet. Prepare your aswers to the followig questios i a plai ASCII text file or
CS / MCS 401 Homework 3 grader solutions
CS / MCS 401 Homework 3 grader solutios assigmet due July 6, 016 writte by Jāis Lazovskis maximum poits: 33 Some questios from CLRS. Questios marked with a asterisk were ot graded. 1 Use the defiitio of
GENERALIZATIONS OF ZECKENDORFS THEOREM. TilVIOTHY J. KELLER Student, Harvey Mudd College, Claremont, California
GENERALIZATIONS OF ZECKENDORFS THEOREM TilVIOTHY J. KELLER Studet, Harvey Mudd College, Claremot, Califoria 91711 The Fiboacci umbers F are defied by the recurrece relatio Fi = F 2 = 1, F = F - + F 0 (
CH5. Discrete Probability Distributions
CH5. Discrete Probabilit Distributios Radom Variables A radom variable is a fuctio or rule that assigs a umerical value to each outcome i the sample space of a radom eperimet. Nomeclature: - Capital letters:
Generating Functions II
Geeratig Fuctios II Misha Lavrov ARML Practice 5/4/2014 Warm-up problems 1. Solve the recursio a +1 = 2a, a 0 = 1 by usig commo sese. 2. Solve the recursio b +1 = 2b + 1, b 0 = 1 by usig commo sese ad
Lecture 4. Random variable and distribution of probability
Itroductio to theory of probability ad statistics Lecture. Radom variable ad distributio of probability dr hab.iż. Katarzya Zarzewsa, prof.agh Katedra Eletroii, AGH e-mail: za@agh.edu.pl http://home.agh.edu.pl/~za
Bertrand 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
0, otherwise. EX = E(X 1 + X n ) = EX j = np and. Var(X j ) = np(1 p). Var(X) = Var(X X n ) =
PROBABILITY MODELS 35 10. Discrete probability distributios I this sectio, we discuss several well-ow discrete probability distributios ad study some of their properties. Some of these distributios, lie
Books 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
Lecture Chapter 6: Convergence of Random Sequences
ECE5: Aalysis of Radom Sigals Fall 6 Lecture Chapter 6: Covergece of Radom Sequeces Dr Salim El Rouayheb Scribe: Abhay Ashutosh Doel, Qibo Zhag, Peiwe Tia, Pegzhe Wag, Lu Liu Radom sequece Defiitio A ifiite
Power Series Expansions of Binomials
Power Series Expasios of Biomials S F Ellermeyer April 0, 008 We are familiar with expadig biomials such as the followig: ( + x) = + x + x ( + x) = + x + x + x ( + x) 4 = + 4x + 6x + 4x + x 4 ( + x) 5
Putnam 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.
It 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
CS 330 Discussion - Probability
CS 330 Discussio - Probability March 24 2017 1 Fudametals of Probability 11 Radom Variables ad Evets A radom variable X is oe whose value is o-determiistic For example, suppose we flip a coi ad set X =
Some Basic Counting Techniques
Some Basic Coutig Techiques Itroductio If A is a oempty subset of a fiite sample space S, the coceptually simplest way to fid the probability of A would be simply to apply the defiitio P (A) = s A p(s);
Homework 1 Solutions. The exercises are from Foundations of Mathematical Analysis by Richard Johnsonbaugh and W.E. Pfaffenberger.
Homewor 1 Solutios Math 171, Sprig 2010 Hery Adams The exercises are from Foudatios of Mathematical Aalysis by Richard Johsobaugh ad W.E. Pfaffeberger. 2.2. Let h : X Y, g : Y Z, ad f : Z W. Prove that
Permutations, Combinations, and the Binomial Theorem
Permutatios, ombiatios, ad the Biomial Theorem Sectio Permutatios outig methods are used to determie the umber of members of a specific set as well as outcomes of a evet. There are may differet ways to
MATH/STAT 352: Lecture 15
MATH/STAT 352: Lecture 15 Sectios 5.2 ad 5.3. Large sample CI for a proportio ad small sample CI for a mea. 1 5.2: Cofidece Iterval for a Proportio Estimatig proportio of successes i a biomial experimet
Math 140A Elementary Analysis Homework Questions 1
Math 14A Elemetary Aalysis Homewor Questios 1 1 Itroductio 1.1 The Set N of Natural Numbers 1 Prove that 1 2 2 2 2 1 ( 1(2 1 for all atural umbers. 2 Prove that 3 11 (8 5 4 2 for all N. 4 (a Guess a formula
Lecture 5. Random variable and distribution of probability
Itroductio to theory of probability ad statistics Lecture 5. Radom variable ad distributio of probability prof. dr hab.iż. Katarzya Zarzewsa Katedra Eletroii, AGH e-mail: za@agh.edu.pl http://home.agh.edu.pl/~za
Combinatorially Thinking
Combiatorially Thiig SIMUW 2008: July 4 25 Jeifer J Qui jjqui@uwashigtoedu Philosophy We wat to costruct our mathematical uderstadig To this ed, our goal is to situate our problems i cocrete coutig cotexts
Discrete 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
Generating Functions. 1 Operations on generating functions
Geeratig Fuctios The geeratig fuctio for a sequece a 0, a,..., a,... is defied to be the power series fx a x. 0 We say that a 0, a,... is the sequece geerated by fx ad a is the coefficiet of x. Example
Introduction To Discrete Mathematics
Itroductio To Discrete Mathematics Review If you put + pigeos i pigeoholes the at least oe hole would have more tha oe pigeo. If (r + objects are put ito boxes, the at least oe of the boxes cotais r or
Objective Mathematics
. If sum of '' terms of a sequece is give by S Tr ( )( ), the 4 5 67 r (d) 4 9 r is equal to : T. Let a, b, c be distict o-zero real umbers such that a, b, c are i harmoic progressio ad a, b, c are i arithmetic
Problems 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
Advanced 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.
Summer High School 2009 Aaron Bertram
Summer High School 009 Aaro Bertram 3 Iductio ad Related Stuff Let s thik for a bit about the followig two familiar equatios: Triagle Number Equatio Square Number Equatio: + + 3 + + = ( + + 3 + 5 + + (
Downloaded from
ocepts ad importat formulae o probability Key cocept: *coditioal probability *properties of coditioal probability *Multiplicatio Theorem o Probablity *idepedet evets *Theorem of Total Probablity *Bayes
Polynomial Equations and Tangents
Polyomial Equatios ad Tagets Jim lowers presetatio to the Mathematical ssociatio of merica MD-DC-V Sectio Meetig 07 pril 9 9:5 am rilliat.org Puzzle Problem appeared i a Facebook post this past witer What
Lecture 10: Mathematical Preliminaries
Lecture : Mathematical Prelimiaries Obective: Reviewig mathematical cocepts ad tools that are frequetly used i the aalysis of algorithms. Lecture # Slide # I this
Binomial distribution questions: formal word problems
Biomial distributio questios: formal word problems For the followig questios, write the iformatio give i a formal way before solvig the problem, somethig like: X = umber of... out of 2, so X B(2, 0.2).
Lecture 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.
Randomized 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
Handout #5. Discrete Random Variables and Probability Distributions
Hadout #5 Title: Foudatios of Ecoometrics Course: Eco 367 Fall/015 Istructor: Dr. I-Mig Chiu Discrete Radom Variables ad Probability Distributios Radom Variable (RV) Cosider the followig experimet: Toss
Week 5-6: The Binomial Coefficients
Wee 5-6: The Biomial Coefficiets March 6, 2018 1 Pascal Formula Theorem 11 (Pascal s Formula For itegers ad such that 1, ( ( ( 1 1 + 1 The umbers ( 2 ( 1 2 ( 2 are triagle umbers, that is, The petago umbers
3.2 Combinations 92 CHAPTER 3. COMBINATORICS
9 CHAPTER. COMBINATORICS (b) Write a computer program to compare Holmes s ad Watso s guessig strategies as follows: fix a total N ad choose 6 itegers radomly betwee ad N. Let m deote the largest of these.
Discrete Probability Functions
Discrete Probability Fuctios Daiel B. Rowe, Ph.D. Professor Departmet of Mathematics, Statistics, ad Computer Sciece Copyright 017 by 1 Outlie Discrete RVs, PMFs, CDFs Discrete Expectatios Discrete Momets
Is mathematics discovered or
996 Chapter 1 Sequeces, Iductio, ad Probability Sectio 1. Objectives Evaluate a biomial coefficiet. Expad a biomial raised to a power. Fid a particular term i a biomial expasio. The Biomial Theorem Galaxies
Topic 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
Probability and Statistics
robability ad Statistics rof. Zheg Zheg Radom Variable A fiite sigle valued fuctio.) that maps the set of all eperimetal outcomes ito the set of real umbers R is a r.v., if the set ) is a evet F ) for
Empirical Distributions
Empirical Distributios A empirical distributio is oe for which each possible evet is assiged a probability derived from experimetal observatio. It is assumed that the evets are idepedet ad the sum of the
Pell and Lucas primes
Notes o Number Theory ad Discrete Mathematics ISSN 30 532 Vol. 2, 205, No. 3, 64 69 Pell ad Lucas primes J. V. Leyedekkers ad A. G. Shao 2 Faculty of Sciece, The Uiversity of Sydey NSW 2006, Australia
UNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST. First Round For all Colorado Students Grades 7-12 November 3, 2007
UNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST First Roud For all Colorado Studets Grades 7- November, 7 The positive itegers are,,, 4, 5, 6, 7, 8, 9,,,,. The Pythagorea Theorem says that a + b =
AMS 216 Stochastic Differential Equations Lecture 02 Copyright by Hongyun Wang, UCSC ( ( )) 2 = E X 2 ( ( )) 2
AMS 216 Stochastic Differetial Equatios Lecture 02 Copyright by Hogyu Wag, UCSC Review of probability theory (Cotiued) Variace: var X We obtai: = E X E( X ) 2 = E( X 2 ) 2E ( X )E X var( X ) = E X 2 Stadard
Probability 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
STAT Homework 1 - Solutions
STAT-36700 Homework 1 - Solutios Fall 018 September 11, 018 This cotais solutios for Homework 1. Please ote that we have icluded several additioal commets ad approaches to the problems to give you better
Topic 8: Expected Values
Topic 8: Jue 6, 20 The simplest summary of quatitative data is the sample mea. Give a radom variable, the correspodig cocept is called the distributioal mea, the epectatio or the epected value. We begi
n 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
Sets and Probabilistic Models
ets ad Probabilistic Models Berli Che Departmet of Computer ciece & Iformatio Egieerig Natioal Taiwa Normal Uiversity Referece: - D. P. Bertsekas, J. N. Tsitsiklis, Itroductio to Probability, ectios 1.1-1.2
4. 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
CS 171 Lecture Outline October 09, 2008
CS 171 Lecture Outlie October 09, 2008 The followig theorem comes very hady whe calculatig the expectatio of a radom variable that takes o o-egative iteger values. Theorem: Let Y be a radom variable that
CS 70 Second Midterm 7 April NAME (1 pt): SID (1 pt): TA (1 pt): Name of Neighbor to your left (1 pt): Name of Neighbor to your right (1 pt):
CS 70 Secod Midter 7 April 2011 NAME (1 pt): SID (1 pt): TA (1 pt): Nae of Neighbor to your left (1 pt): Nae of Neighbor to your right (1 pt): Istructios: This is a closed book, closed calculator, closed
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
(ii) Two-permutations of {a, b, c}. Answer. (B) P (3, 3) = 3! (C) 3! = 6, and there are 6 items in (A). ... Answer.
SOLUTIONS Homewor 5 Due /6/19 Exercise. (a Cosider the set {a, b, c}. For each of the followig, (A list the objects described, (B give a formula that tells you how may you should have listed, ad (C verify
MATH 304: MIDTERM EXAM SOLUTIONS
MATH 304: MIDTERM EXAM SOLUTIONS [The problems are each worth five poits, except for problem 8, which is worth 8 poits. Thus there are 43 possible poits.] 1. Use the Euclidea algorithm to fid the greatest
Principle of Strong Induction
Strog Iductio Pricile of Strog Iductio Let P() be a statemet about the th iteger. If the followig hyotheses hold: i. P(1) is True. ii. The statemet P(1) P(2) P() P( +1) is True for all 1. The we ca coclude
CSE 21 Mathematics for
CSE 2 Mathematics for Algorithm ad System Aalysis Summer, 2005 Outlie What a geeratig fuctio is How to create a geeratig fuctio to model a problem Fidig the desired coefficiet Partitios Expoetial geeratig
0 1 sum= sum= sum= sum= sum= sum= sum=64
Biomial Coefficiets I how may ways ca we choose elemets from a elemet set? There are choices for the first elemet, - for the secod,..., dow to - + for the th, yieldig *(-)*...*(-+). So there are 4*3=2
An Introduction to Randomized Algorithms
A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis
1. By using truth tables prove that, for all statements P and Q, the statement
Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3
Lecture 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,
Different kinds of Mathematical Induction
Differet ids of Mathematical Iductio () Mathematical Iductio Give A N, [ A (a A a A)] A N () (First) Priciple of Mathematical Iductio Let P() be a propositio (ope setece), if we put A { : N p() is true}
Econ 325: Introduction to Empirical Economics
Eco 35: Itroductio to Empirical Ecoomics Lecture 3 Discrete Radom Variables ad Probability Distributios Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-1 4.1 Itroductio to Probability
CS166 Handout 02 Spring 2018 April 3, 2018 Mathematical Terms and Identities
CS166 Hadout 02 Sprig 2018 April 3, 2018 Mathematical Terms ad Idetities Thaks to Ady Nguye ad Julie Tibshirai for their advice o this hadout. This hadout covers mathematical otatio ad idetities that may
Math 4107: Abstract Algebra I Fall Webwork Assignment1-Groups (5 parts/problems) Solutions are on Webwork.
Math 4107: Abstract Algebra I Fall 2017 Assigmet 1 Solutios 1. Webwork Assigmet1-Groups 5 parts/problems) Solutios are o Webwork. 2. Webwork Assigmet1-Subgroups 5 parts/problems) Solutios are o Webwork.
Approximations 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.
CSE 1400 Applied Discrete Mathematics Number Theory and Proofs
CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of
Section 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
Math 4400/6400 Homework #7 solutions
MATH 4400 problems. Math 4400/6400 Homewor #7 solutios 1. Let p be a prime umber. Show that the order of 1 + p modulo p 2 is exactly p. Hit: Expad (1 + p) p by the biomial theorem, ad recall from MATH
REVIEW FOR CHAPTER 1
REVIEW FOR CHAPTER 1 A short summary: I this chapter you helped develop some basic coutig priciples. I particular, the uses of ordered pairs (The Product Priciple), fuctios, ad set partitios (The Sum Priciple)
MT5821 Advanced Combinatorics
MT5821 Advaced Combiatorics 9 Set partitios ad permutatios It could be said that the mai objects of iterest i combiatorics are subsets, partitios ad permutatios of a fiite set. We have spet some time coutig