Conditional Probability. Given an event M with non zero probability and the condition P( M ) > 0, Independent Events P A P (AB) B P (B)
|
|
- Miles King
- 5 years ago
- Views:
Transcription
1 Coditioal robability Give a evet M with o zero probability ad the coditio ( M ) > 0, ( / M ) ( M ) ( M ) Idepedet Evets () ( ) ( ) () ( ) ( ) ( )
2 Examples ) Let items be chose at radom from a lot cotaiig items of which 4 are defective. If {both items are defective} ad {both items are o-defective}, fid () ad ) (). S ca occur i 66 ways, ca occur i ways, ca occur i 8 ways, The 8 4 ( ). 66 The probability that at least oe item is defective(c) {at least oe item is defective} is the complemet of ; that is, C c. Thus by Theorem., C p( C) ( ) ( ) 4 9 Note: The odds that a evet with probability p occurs is defied to be the ratio p: (-p). Thus 4 the odds that at least oe item is defective is or 9:4 which is read 9 to 4. : ) Three horses, ad C are i a race; is twice as likely to wi as ad is twice as likely to wi as C. What are their respective probabilities of wiig, i.e. ( ), ( ) ad ( C )? ccordigly, () 4p 4, 7 () p, 7 (C) p 7 p + p + 4p or 7p or p 7
3 robabilities o the Real Lie ssume ζ is the set of positive real umber t i 0 t t, for ay 0 t t, z t 0 α (t) dt where α (t) 0 prob. of a evet { } Note: t t 0,. evet is { } t t t 0 t t - 0 t < t zt α (t) dt t
4 Examples oits a ad b are selected at radom such that b 0 ad 0 a as show below. Fid the probability p that the distace d betwee a ad b is greater tha. d a - b - b 0 a The sample space S X p area of ( ) area of S 6
5 Total robability Give mutually exclusive evets,,...,, ad S Theorem of Total robability: ( ) ( ) ι ι ι ( ) ( ) ( ) ( ) ( ) ( ) N L i i
6 ayes Theorum i posterieor i β i i j probability, ( ) i ( ) i ( ) prior probability i I geeral we ca state the followig state sample { sample state} { state} { sample } { state} state
7 Example: Samplig problem 80 g items ad 0 d items. O the secod draw, we have, 0 80 defect o st draw ( ), ( ) ( ) 9 defect o d draw, 99 ( ) ( ) ( ) + ( )
8 Examples Three Maufactures Supplier #---50% Supplier #---0% Supplier #---0% 95% compoets pass 90% compoets pass 80% compoets pass Give that a compoet passes ispectio, fid the probability that it has bee supplied by each supplier. Solutio: Let E i evet that a particular compoet is supplied by the ith supplier. Let {ay oe compoet passes ispectio tests.} E i [ ] E E EL ( E ) E ( 0.50)( 0.95) ( 0.50)( 0.95) + ( 0.0)( 0.90) + ( 0.0)( 0.80) 0.55 E E 0.97, 0.78
9 Example: O the spot check of the breakdow of a machie idicated that it is caused by overheatig of the motor. ast records show (correct)0.75 (correct/overheat)0.9 (overheat)0.8 a) Fid the probability that a breakdow is caused by the overheatig of the motor, give that the o the spot check is correct, idicatig that such is the case. b) Fid the probability of makig a correct aalysis from a o the spot check, give that a breakdow is ot caused by the overheatig of the motor. Let (correct aalysis of the cause of the breakdow is made through o the spot checks) E(breakdow is caused by overheatig) {}robability of correct aalysis through o the spot checks. ( E) a) [ ] [ ] ( 4 )( 9 ) E E [ ] ( 0 ) 4 Let E evet that breakdow is ot caused by overheatig. ( E' ) ( E) ; 5 [ D ] robability of makig correct o - the - spot aalysis give b) E' failure ot by overheatig ( ) ( ) ( ) + ( ') ( E E E E' ) ' ' 0. 5 E E robability of makig the correct aalysis from a o spot check, give that a breakdow is ot caused by overheatig. ( )( ) ( ) ( ) 5
10 Example Oe coi i a roll of cois has two heads. The others are fair cois. coi is selected at radom ad tossed m times. What is the probability that the coi tossed is two-headed, give that all m tosses are head. How large must m be for this probability to be greater tha ½? Solutio: Let evet two-headed coi selected. Let evet all m throws are head. a) / N-/ ( ( ) ) m ( ) (.5) m m (-(.5) ) + ( ) 0. 5 b ) For > l( ) l
11 Example The Simpso test is used by Compay to test 000 employees. Maagemet estimates that % of employees use drugs. The test is 98% accurate whe someoe is actually doig drugs, ad is % false positive. a) perso has a positive test. What is the probability that the perso truly uses drugs? b) If a perso tests egative, what is the probability he uses drugs? Solutios: ( D + ) + ( ) T ( D) (.0)(.9ε) a) D.48 + T ( T ) (. )(.9ε) + (.99)(.0) ( ).99.0 ND D ( ) T - T + T - T T D(0).0*.0 b) d D T ( T ).0*.0*.99*.97 +
12 Example game show cotestat selects oe of three curtais that he thiks hides a car. The other two curtais hide goats. fter he chooses, host opes oe of the remaiig curtais at radom to reveal a goat. Should the cotestat stay with his choice, or should he switch to the remaiig curtai? Geeralize to curtais, oe car ad - goats, where oe curtai is selected by the cotestat ad - of the - emaiig curtais are opeed by the host to reveal goats. Solutio: Let - (car)/(choosig car) (sg/hc)(shows goat/havig car) (goat)/(choosig goat) (hg/sg)(have goat/shows goat) (hc/sg)(havig a car/shows goat) (sh/hg)(shows goat/have goat) Usig aye s Rule sg ( hc) sg ( hg) hc hc ; hg hg sg ( sg) sg ( sg) ssumig the host selects a curtai at radom, the. (sg)(sg/car)(car)+(sg/goat)(goat) * + * ( if at radom, if ot, the prob. ) The: ( hc sg) * * hg sg We see it makes o differece if he chages or ot.
13 If the host kows where the goats are, host deliberately turs over curtai with goat sg hc hc sg ( hc) + ( hg ) * + * hg sg sg hg * ( )( ) + ( )( ) * * Hece, if the game is rigged, it pays to chage the selectio. You ca double your chace.
14 Whe there are curtais, sg ( hc) hc hc sg * sg We have oe car ad - goats. The,, sg sg or if game is rigged, hc hg - (hc) /, (hg) ( sg) ( sg / hc) ( hc) + ( sg / hg) ( hg) x/ + ( - )/( -)x( -)/ ( -)/ { hc / sg} Or, if the game is rigged, ( ) x hc sg { / }. { sg / hc} { hc} { sg} ( this is equal to ).
15 Cotiuig ( hg / sg ) ( sg / hg) ( hg) ( sg) - x or, if fixed * We see agai, there is a advatage to switchig curtais if you assume that the game host kows which curtai is hidig the car.
16 Idepedet Evets If ad are idepedet, the _ ad _ ad ad roof of (a) ( ) ( ) ad ( ) ( ) So Thus ( ) ( ) ( ) ( ) + ( ) ( ) ( ) + ( ) ( ) ( ) ( ) ( ) ( ) [ ( ) ] ( ) ( ) ( ) ( ) ( ) QED
17 roof of (b) ( ) ( ) ( ) ( ) + ( ) ( ) ( ) + ( ) ( ) ()[ - ()] ()() QED roof of (c) Usig DeMorga s Theorem, ( ) ( ) ( ) ( ) + ( ) ( ) ( ) ( ) [ ( ) ] ( ( ) )( ( ) ) ( ) ( ) QED
18 Defiitio ( ) sequece of evets,,,..., is said to be mutuallyidepede t if for each set for distict idicies, i,i,...i that are elemets of {,,..., } we have : ( i i... i ) ( i )... ( i ) κ Κ Κ NOTE : Ca have ( C) ( ) ( ) ( C) but ( ) ( ) ( ) Example: Experimet of tossig two dice. S {( i, j) : i, 6} j Let: First die results i,, or First die results i,4,5 C The sum of the two faces is 9 ( ), ( ), ( C ) 9 C 6 6 ut, ( ) ( ) ( ) 6 6 ot mutually idepedet. ( C) ( ) ( ) ( C) 4 {(, ),(,),(, ), (,4),(,5),(,6) } {(,6) } C {(,6),( 4,5),( 5,4) } C {(,6) }
19 Example bag cotais four marbles umbered,,, ad 4. Defie: - Draw marbles ad - Draw marbles ad C - Draw marbles ad 4 re,, C mutually idepedet? Solutio: robability of drawig ay two marbles is /4 +/4/. robability of drawig ay two first, ad the ay other two (assumig replacemet) is always (.5)x(.5) ()()()(C) etc, etc Hece, the evets are mutually idepedet.
20 Example Experimet of tossig two dice. S {( i, j) : i, 6} j Let: First die results i,, or First die results i,4,5 C The sum of the two faces is 9 C C {(, ), (,),(,),(,4), (,5 ), (,6) } {(,6) } {(,6), ( 4,5 ), ( 5,4) } C {(,6) } Thus,, ad C are ot mutually idepedet. 6 ut, ( ) ( ) ( ) 6 6 ( ), ( ), ( C) 9 4
Chapter 6 Conditional Probability
Lecture Notes o robability Coditioal robability 6. Suppose RE a radom experimet S sample space C subset of S φ (i.e. (C > 0 A ay evet Give that C must occur, the the probability that A happe is the coditioal
More informationCS 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 =
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 informationExpectation and Variance of a random variable
Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio
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 informationHomework 5 Solutions
Homework 5 Solutios p329 # 12 No. To estimate the chace you eed the expected value ad stadard error. To do get the expected value you eed the average of the box ad to get the stadard error you eed the
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 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 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 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 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 informationDownloaded 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
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 information1 Review of Probability & Statistics
1 Review of Probability & Statistics a. I a group of 000 people, it has bee reported that there are: 61 smokers 670 over 5 960 people who imbibe (drik alcohol) 86 smokers who imbibe 90 imbibers over 5
More informationProbability 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
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 informationf 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)
More informationCS / 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
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 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 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 informationCS 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
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 informationChapter 6 Sampling Distributions
Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to
More informationf 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)
More informationPRACTICE PROBLEMS FOR THE FINAL
PRACTICE PROBLEMS FOR THE FINAL Math 36Q Fall 25 Professor Hoh Below is a list of practice questios for the Fial Exam. I would suggest also goig over the practice problems ad exams for Exam ad Exam 2 to
More informationAs 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
More information1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable
More informationIntroduction 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
More informationQuick Review of Probability
Quick Review of Probability Berli Che Departmet of Computer Sciece & Iformatio Egieerig Natioal Taiwa Normal Uiversity Refereces: 1. W. Navidi. Statistics for Egieerig ad Scietists. Chapter 2 & Teachig
More informationQuick Review of Probability
Quick Review of Probability Berli Che Departmet of Computer Sciece & Iformatio Egieerig Natioal Taiwa Normal Uiversity Refereces: 1. W. Navidi. Statistics for Egieerig ad Scietists. Chapter & Teachig Material.
More informationThe variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.
SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample
More informationChapter Vectors
Chapter 4. Vectors fter readig this chapter you should be able to:. defie a vector. add ad subtract vectors. fid liear combiatios of vectors ad their relatioship to a set of equatios 4. explai what it
More informationLectures 1 5 Probability Models
Lectures 1 5 Probability Models Aalogy with Geometry: abstract model for chace pheomea Laguage ad Symbols of Chace Experimets: Sample space S, cosistig of all possible outcomes (elemets e, f,..., evets
More informationSome 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
More informationRandom 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 informationBinomial 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).
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 informationPROBABILITY. Note : Probability of occurrence of an event A is denoted by P(A).
J-Mathematics PROBABILITY INTRODUCTION : The theory of probability has bee origiated from the game of chace ad gamblig. I old days, gamblers used to gamble i a gamblig house with a die to wi the amout
More informationPRACTICE PROBLEMS FOR THE FINAL
PRACTICE PROBLEMS FOR THE FINAL Math 36Q Sprig 25 Professor Hoh Below is a list of practice questios for the Fial Exam. I would suggest also goig over the practice problems ad exams for Exam ad Exam 2
More informationWhat 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
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 informationAverage Case Complexity
Probability Applicatios Aalysis of Algorithms Average Case Complexity Mote Carlo Methods Spam Filters Probability Distributio Basic Cocepts S = {a 1,, a } = fiite set of outcomes = sample space p :S [0,1]
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 informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More informationLecture 1 Probability and Statistics
Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark
More informationLecture 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
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 information0, 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
More informationBinomial Distribution
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible
More informationCS 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
More informationDesign and Analysis of Algorithms
Desig ad Aalysis of Algorithms Probabilistic aalysis ad Radomized algorithms Referece: CLRS Chapter 5 Topics: Hirig problem Idicatio radom variables Radomized algorithms Huo Hogwei 1 The hirig problem
More informationSets 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
More informationGenerating 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
More informationChapter 5: Hypothesis testing
Slide 5. Chapter 5: Hypothesis testig Hypothesis testig is about makig decisios Is a hypothesis true or false? Are wome paid less, o average, tha me? Barrow, Statistics for Ecoomics, Accoutig ad Busiess
More informationConsider unordered sample of size r. This sample can be used to make r! Ordered samples (r! permutations). unordered sample
Uodeed Samples without Replacemet oside populatio of elemets a a... a. y uodeed aagemet of elemets is called a uodeed sample of size. Two uodeed samples ae diffeet oly if oe cotais a elemet ot cotaied
More informationREAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS
REAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS 18th Feb, 016 Defiitio (Lipschitz fuctio). A fuctio f : R R is said to be Lipschitz if there exists a positive real umber c such that for ay x, y i the domai
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationChapter 0. Review of set theory. 0.1 Sets
Chapter 0 Review of set theory Set theory plays a cetral role i the theory of probability. Thus, we will ope this course with a quick review of those otios of set theory which will be used repeatedly.
More informationProbability Refresher and Cycle Analysis. Spring 2018 CS 438 Staff, University of Illinois 1
Probability Refresher ad Cycle Aalysis Sprig 2018 CS 438 Staff, Uiversity of Illiois 1 A Quick Probability Refresher A radom variable, X, ca take o a umber of differet possible values Example: the umber
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 informationInduction: Solutions
Writig Proofs Misha Lavrov Iductio: Solutios Wester PA ARML Practice March 6, 206. Prove that a 2 2 chessboard with ay oe square removed ca always be covered by shaped tiles. Solutio : We iduct o. For
More informationMA131 - Analysis 1. Workbook 2 Sequences I
MA3 - Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................
More informationAssignment ( ) 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:
More informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
More informationLecture 5: Parametric Hypothesis Testing: Comparing Means. GENOME 560, Spring 2016 Doug Fowler, GS
Lecture 5: Parametric Hypothesis Testig: Comparig Meas GENOME 560, Sprig 2016 Doug Fowler, GS (dfowler@uw.edu) 1 Review from last week What is a cofidece iterval? 2 Review from last week What is a cofidece
More informationThe 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
More informationsin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =
60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece
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 information(7 One- and Two-Sample Estimation Problem )
34 Stat Lecture Notes (7 Oe- ad Two-Sample Estimatio Problem ) ( Book*: Chapter 8,pg65) Probability& Statistics for Egieers & Scietists By Walpole, Myers, Myers, Ye Estimatio 1 ) ( ˆ S P i i Poit estimate:
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More informationReal Variables II Homework Set #5
Real Variables II Homework Set #5 Name: Due Friday /0 by 4pm (at GOS-4) Istructios: () Attach this page to the frot of your homework assigmet you tur i (or write each problem before your solutio). () Please
More informationAn 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
More informationUC Berkeley Department of Electrical Engineering and Computer Sciences. EE126: Probability and Random Processes
UC Berkeley Departmet of Electrical Egieerig ad Computer Scieces EE26: Probability ad Radom Processes Problem Set Fall 208 Issued: Thursday, August 23, 208 Due: Wedesday, August 29, 207 Problem. (i Sho
More informationDS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10
DS 00: Priciples ad Techiques of Data Sciece Date: April 3, 208 Name: Hypothesis Testig Discussio #0. Defie these terms below as they relate to hypothesis testig. a) Data Geeratio Model: Solutio: A set
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 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 informationAMS570 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)
More informationUNIVERSITY 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 =
More informationApril 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE
April 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE TERRY SOO Abstract These otes are adapted from whe I taught Math 526 ad meat to give a quick itroductio to cofidece
More informationDiscrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand Final Solutions
CS 70 Discrete Mathematics ad Probability Theory Fall 2016 Seshia ad Walrad Fial Solutios CS 70, Fall 2016, Fial Solutios 1 1 TRUE or FALSE?: 2x8=16 poits Clearly put your aswers i the aswer box o the
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 informationAlgebra of Least Squares
October 19, 2018 Algebra of Least Squares Geometry of Least Squares Recall that out data is like a table [Y X] where Y collects observatios o the depedet variable Y ad X collects observatios o the k-dimesioal
More informationEXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY
EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA, 016 MODULE : Statistical Iferece Time allowed: Three hours Cadidates should aswer FIVE questios. All questios carry equal marks. The umber
More informationLecture 1 Probability and Statistics
Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark
More informationIntroduction to Probability. Ariel Yadin. Lecture 7
Itroductio to Probability Ariel Yadi Lecture 7 1. Idepedece Revisited 1.1. Some remiders. Let (Ω, F, P) be a probability space. Give a collectio of subsets K F, recall that the σ-algebra geerated by K,
More informationExploring Basic Probability V. Krishnan Presentation to CACT - September 28, 2001
Eplorig Basic Probability V. Krisha Presetatio to CACT - September 8, Abstract I this talk we will discuss some basic ideas o Probability like idepedece, coditioal probability, Bayes' theorem, aspects
More informationLinearly Independent Sets, Bases. Review. Remarks. A set of vectors,,, in a vector space is said to be linearly independent if the vector equation
Liearly Idepedet Sets Bases p p c c p Review { v v vp} A set of vectors i a vector space is said to be liearly idepedet if the vector equatio cv + c v + + c has oly the trivial solutio = = { v v vp} The
More informationKeywords: Last-Success-Problem; Odds-Theorem; Optimal stopping; Optimal threshold AMS 2010 Mathematics Subject Classification 60G40, 62L15
CONCERNING AN ADVERSARIAL VERSION OF THE LAST-SUCCESS-PROBLEM arxiv:8.0538v [math.pr] 3 Dec 08 J.M. GRAU RIBAS Abstract. There are idepedet Beroulli radom variables with parameters p i that are observed
More informationDiscrete Mathematics for CS Spring 2008 David Wagner Note 22
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig
More informationLOGO. Chapter 2 Discrete Random Variables(R.V) Part1. iugaza2010.blogspot.com
1 LOGO Chapter 2 Discrete Radom Variables(R.V) Part1 iugaza2010.blogspot.com 2.1 Radom Variables A radom variable over a sample space is a fuctio that maps every sample poit (i.e. outcome) to a real umber.
More informationData Analysis and Statistical Methods Statistics 651
Data Aalysis ad Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasii/teachig.html Suhasii Subba Rao Review of testig: Example The admistrator of a ursig home wats to do a time ad motio
More informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More informationSets and Probabilistic Models
ets ad Probabilistic Models Berli Che Departmet of Computer ciece & Iformatio Egieerig Natioal Taiwa Normal iversity Referece: - D. P. Bertsekas, J. N. Tsitsiklis, Itroductio to Probability, ectios 1.1-1.2
More informationCSE 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
More informationThe Boolean Ring of Intervals
MATH 532 Lebesgue Measure Dr. Neal, WKU We ow shall apply the results obtaied about outer measure to the legth measure o the real lie. Throughout, our space X will be the set of real umbers R. Whe ecessary,
More informationCH5. 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:
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
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 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 information