Probability and Statistics
|
|
- Thomas Collins
- 5 years ago
- Views:
Transcription
1 robability ad Statistics rof. Zheg Zheg
2 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 every i R, ad the prob. of the evets {= } {=- } are 0. is a r.v, if B) F where B represets semi-defiite itervals of the form { a} ad all other sets that ca be costructed from these sets by performig the set operatios of uio, itersectio ad egatio ay umber of times. ) A B R
3 if is a r.v, the ) a b is also a evet { a} b are evets, c a a is a evet, Thus a b { a b} is a evet is also a evet a a { a } 3
4 robability Distributio Fuctio DF) Deote ) F ) 0 F ) is said to the robability Distributio Fuctio DF) associated with the r.v. The subscript is to idetify the r.v. if g) is a DF, the it is odecreasig, rightcotiuous, e.g. i) ii) if the. g ), g ) 0,, g ) g ), iii) g ) g ), for all. 4
5 From the earlier defiitio of F ), we have i) F ) ) ) ad F ) ) 0. ) ii) If, the the subset, ), ). ), Cosequetly the evet ) sice ) implies ). As a result F ) ) F ), ) implyig that the probability distributio fuctio is oegative ad mootoe odecreasig. iii) Let, ad cosider the evet A ). sice ) ) ), 5
6 usig mutually eclusive property of evets we get ) F ) F ). A ) But A A A, ad hece Thus lim A A ad hece lim A ) 0. lim A ) lim ) 0. But lim, the right limit of, ad hece F F ) F ) F ), i.e., F ) is right-cotiuous, justifyig all properties of a distributio fuctio. 6
7 Additioal roperties of a DF iv) If F 0 ) 0 for some, the This follows, sice F 0 ) ) 0 0 implies ) 0 is the ull set, ad for ay 0, ) will be a subset of the ull set. v) We have ) ), ad sice the two evets are mutually eclusive, the above follows. vi) ) F ). ) 0,. 0 0 The evets ) ad { ) } are mutually eclusive ad their uio represets the evet ) F ) F ) F ),.. 7
8 vii) ) F ) F ). Let, 0, ad. Sice or lim 0 ) F ) lim F ), 0 ) F ) F ). F 0 ), ), the limit of F as 0 from the right always eists ad equals F 0 ). However the left limit value F 0 ) eed ot equal F 0 ). Thus F ) eed ot be cotiuous from the left. At a discotiuity poit of the distributio, the left ad right limits are differet, ad from above ) F ) F )
9 Thus the oly discotiuities of a distributio fuctio are of the jump type, ad occur at poits 0 where it is satisfied. These poits ca always be eumerated as a sequece, ad moreover they are at most coutable i umber. Eample : is a r.v such that ) c,. Fid Solutio: For c, ), so that F ) 0, ad for c, ), so that. F ) ). F ) F F ) c Eample : Toss a coi. H,T. Suppose the r.v is such that T ) 0, H ). Fid ). F 9
10 Solutio: For 0, ), so that 0,, ) ) T, so that F ) T p, ) H, T, so that F ). Fig. 3) is said to be a cotiuous-type r.v if its distributio fuctio F ) is cotiuous. I that case F ) F all, ad we get 0. If F ) is costat ecept for a fiite umber of jump discotiuitiespiece-wise costat; step-type), the is said to be a discrete-type r.v. If i is such a discotiuity poit, the p F ) F ). i q F ) i i i F 0. ) for 0
11 From the Fig., at a poit of discotiuity we get c F c) F c ) 0. ad from the Fig., 0 F 0) F 0 ) q 0 q. Eample 3. A fair coi is tossed twice, ad let the r.v represet the umber of heads. Fid F ). Solutio: I this case HH, HT, TH, TT, ad 0 0,,,, HH ), HT ), TH ), TT ) ) ) TT F ) TT ) TT, HT, TH F ) TT, HT, TH ) F F ) ) 0,. T 0. ) T ) 4, 3 4,
12 From the Fig.3, robability desity fuctio p.d.f) The derivative of the distributio fuctio F ) is called the probability desity fuctio f ) of the r.v. Thus Sice df d ) ) 3 / 4 / 4 /. 3/ 4 / 4 ) F ) f F F df d ). from the mootoe-odecreasig ature of ) lim 0 F ) F ) 0, F ),
13 it follows that f ) 0 for all. f ) will be a cotiuous fuctio, if is a cotiuous type r.v. However, if is a discrete type r.v as i the above, the f ) its p.d.f has the geeral form Fig. 5) f ) p where i represet the jump-discotiuity poits i F As Fig. 5 shows f ) represets a collectio of positive discrete masses, ad it is ow as the probability mass fuctio p.m.f ) i the discrete case. We also obtai by itegratio Sice it yields F ) i F, i i ), ) f u ) du f ) d,. Fig. 5 p i i ). 3
14 which justifies its ame as the desity fuctio. Further, we also get Fig. 6b). ) F ) F ) f ) d Thus the area uder f ) i the iterval, ) represets the probability. F ) f ) a) b) Fig. 6 Ofte, r.vs are referred by their specific desity fuctios - both i the cotiuous ad discrete cases - ad i what follows we shall list a umber of them i each category. 4
15 Cotiuous-type radom variables. Normal Gaussia): is said to be ormal or Gaussia r.v, if f ) ) This is a bell shaped curve, symmetric aroud the parameter, ad its distributio fuctio is give by F ) ) y e y / where G ) e dy is ofte tabulated. Sice f ) depeds o two parameters ad, the otatio is ofte used. f ) e / / dy. G, N, ) Fig. 7 5
16 . Uiform: U a, b), a b, if Fig. 8) f ) b 0, a, a otherwise. b, 3. Epoetial: ) if Fig. 9) f ) e 0, /, 0, otherwise. b a f ) a Fig. 8 b f ) Fig. 9 6
17 4. Gamma: G, ) if 0, 0) Fig. 0) f ) ) 0, If a iteger e /, otherwise. 0, ) )!. f f ) Fig. 0 ) 5. Beta: a, b) if a 0, b 0) Fig. ) f ) a, b ) 0, a ) b, otherwise. 0, 0 Fig. where the Beta fuctio a, b) is defied as a b a, b ) u u ) du 0. 7
18 6. Chi-Square: ), if Fig. ) f ) 0, / / e / / ) otherwise. 0, Note that ) is the same as Gamma /, )., f ) Fig. 7. Rayleigh: R ), if Fig. 3) f ) e 0, /, otherwise. 0, f ) Fig. 3 8
19 Discrete-type radom variables. Beroulli: taes the values 0,), ad 0) q, ) p.. Biomial: if Fig. 7) B, p), ) p q, 0,,,, 3. oisso: ), if Fig. 8) ) e, 0,,,,.! ) ). Fig. 7 Fig. 8 9
20 4. Hypergeometric: ) m Nm N 5. Geometric: g p ) if, ma0, m N) mi m, ) ) pq, 0,,,,, q p. 6. Negative Biomial: ~ NB r, p), if r r ) p q, r, r,. r 7. Discrete-Uiform: ),,,, N. N 0
21 olya s distributio Icludes both biomial ad hypergeometric as special cases. A bo cotais a white balls ad b blac balls. A ball is draw at radom, ad it is replaced alog with c balls of the same color. If represets the umber of white balls draw i such draws, 0,,,,, fid the probability mass fuctio of. Solutio: Cosider the specific sequece of draws where white balls are first draw, followed by blac balls. The probability of drawig successive white balls is give by p W a a c a c a ) c a b a b c a b c a b ) c Similarly the probability of drawig white balls
22 followed by blac balls is give by b b c b ) c p p w a b c a b ) c a b ) c aic b jc abic ab j) c i0 j0. Iterestigly, p i above also represets the probability of drawig white balls ad ) blac balls i ay other specific order i.e., The same set of umerator ad deomiator terms i above cotribute to all other sequeces as well.) But there are such distict mutually eclusive sequeces ad summig over all of them, we obtai the olya distributio probability of gettig white balls i draws) to be a ic b jc ) p, 0,,,,. i0 j0 abic ab j) c
23 Both biomial distributio as well as the hypergeometric distributio are its special cases. For eample if draws are doe with replacemet, the c = 0 ad it simplifies to the biomial distributio where ) p q, 0,,,, a b p, q p. a b a b Similarly if the draws are coducted without replacemet, The c =, ad it gives ) aa ) a) a) bb ) b) ab) ab) ab ) ab) ab) 3
24 ! a! ab)! b! ab)! )! )! a)! ab)! b)! ab)! which represets the hypergeometric distributio. Fially c = + gives replacemets are doubled) ) a )! ab)! b )! ab)! a)! ab )! b)! ab)! a b ab =. we shall refer it as olya s + distributio. the geeral olya distributio has bee used to study the spread of cotagious diseases epidemic modelig). a b ab 4
25 5 Let represet a Biomial r.v, the Sice the biomial coefficiet grows quite rapidly with, it is difficult to compute it for large. I this cotet, two approimatios are etremely useful. The Normal Approimatio Demoivre-Laplace Theorem) Suppose with p held fied. The for i the eighborhood of p, we ca approimate. ) q p! )!! pq Biomial Radom Variable Approimatios
26 p q p ) / pq pq Thus if ad are withi or aroud the eighborhood of the iterval p pq, p pq, we ca approimate the summatio by a itegratio. I that case it reduces to where e pq e p) / pq y / d p pq, p pq.. e dy, We ca epress it i terms of the ormalized itegral erf ) erf ) that has bee tabulated etesively See Table ). 0 e y / dy 6
27 For eample, if ad are both positive,we obtai erf ) erf ). Eample : A fair coi is tossed 5,000 times. Fid the probability that the umber of heads is betwee,475 to,55. Solutio: We eed,475,55 ). Here is large so that we ca use the ormal approimatio. I this case p so that p,500 ad 35. Sice p pq ad pq,535, the approimatio is valid for, ad,55. Thus Here pq,465, p y / e 5 p p, pq pq 5 7. dy. 7,
28 8 erf) erf) erf) erf) ) ) erf 0 / G dy e y Table
29 Sice 0, from Fig. b), the above probability is give by,475,55 erf ) erf ) erf ) erf erf where we have used Table a) The oisso Approimatio e 0, 0 / , 7 erf0.7) Fig. As we have metioed earlier, for large, the Gaussia approimatio of a biomial r.v is valid oly if p is fied, i.e., oly if p ad pq. what if p is small, or if it does ot icrease with? b) e 0, 0 / ) 9
30 Obviously that is the case if, for eample, 0 as such that is a fied umber. p p, May radom pheomea i ature i fact follow this patter. Total umber of calls o a telephoe lie, claims i a isurace compay etc. ted to follow this type of behavior. Cosider radom arrivals such as telephoe calls over a lie. Let represet the total umber of calls i the iterval 0, ). From our eperiece, as T we have so that we may assume T. Cosider a small iterval of duratio as i Fig.. If there is oly a sigle call comig i, the probability p of that sigle call occurrig i that iterval must deped o its relative size with respect to T. T 0 T Fig. 30
31 Hece we may assume p. Note that as T p 0 T. However i this case p T is a costat, T ad the ormal approimatio is ivalid here. Suppose the iterval i Fig. is of iterest to us. A call iside that iterval is a success H), whereas oe outside is a failure T ). This is equivalet to the coi tossig situatio, ad hece the probability ) of obtaiig calls i ay order) i a iterval of duratio is give by the biomial p.m.f. Thus )! )!! ad here as, p 0 such that p. It is easy to obtai a ecellet approimatio i that situatio. To see this, rewrite it as p p), 3
32 3. ) / ) /! ) /! ) ) ) ) p p,! ) lim 0,, e p p sice the fiite products as well as ted to uity as ad The right side of it represets the oisso p.m.f ad the oisso approimatio to the biomial r.v is valid i situatios where the biomial r.v parameters ad p diverge to two etremes such that their product p is a costat.,. lim e 0), p Thus
33 Eample : Wiig a Lottery: Suppose two millio lottery ticets are issued with 00 wiig ticets amog them. a) If a perso purchases 00 ticets, what is the probability of wiig? b) How may ticets should oe buy to be 95% cofidet of havig a wiig ticet? Solutio: The probability of buyig a wiig ticet p No. of wiig ticets Total o. of ticets 0 Here 00, ad the umber of wiig ticets i the purchased ticets has a approimate oisso distributio with parameter Thus p 00 ad a) robability of wiig ) e 5! 0, ) 0) e
34 b) I this case we eed ) ) e 0.95 implies l 0 But p or 60,000. Thus oe eeds to buy about 60,000 ticets to be 95% cofidet of havig a wiig ticet! Eample 3: A space craft has 0 5 compoets The probability of ay oe compoet beig defective is 5 0 p 0). The missio will be i dager if five or more compoets become defective. Fid the probability of such a evet. Solutio: Here is large ad p is small, ad hece oisso approimatio is valid. Thus p 00, ad the desired probability is give by 4 4 5) 4) e e e ! ! 3. 34,
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
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 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 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 informationB Supplemental Notes 2 Hypergeometric, Binomial, Poisson and Multinomial Random Variables and Borel Sets
B671-672 Supplemetal otes 2 Hypergeometric, Biomial, Poisso ad Multiomial Radom Variables ad Borel Sets 1 Biomial Approximatio to the Hypergeometric Recall that the Hypergeometric istributio is fx = x
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 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 information4. Basic probability theory
Cotets Basic cocepts Discrete radom variables Discrete distributios (br distributios) Cotiuous radom variables Cotiuous distributios (time distributios) Other radom variables Lect04.ppt S-38.45 - Itroductio
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 informationSingular Continuous Measures by Michael Pejic 5/14/10
Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σ-algebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable
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 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 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 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 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 informationDiscrete Mathematics and Probability Theory Summer 2014 James Cook Note 15
CS 70 Discrete Mathematics ad Probability Theory Summer 2014 James Cook Note 15 Some Importat Distributios I this ote we will itroduce three importat probability distributios that are widely used to model
More informationLecture 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
More informationIntegrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number
MATH 532 Itegrable Fuctios Dr. Neal, WKU We ow shall defie what it meas for a measurable fuctio to be itegrable, show that all itegral properties of simple fuctios still hold, ad the give some coditios
More informationDiscrete Mathematics and Probability Theory Spring 2012 Alistair Sinclair Note 15
CS 70 Discrete Mathematics ad Probability Theory Sprig 2012 Alistair Siclair Note 15 Some Importat Distributios The first importat distributio we leared about i the last Lecture Note is the biomial distributio
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 informationLast time: Moments of the Poisson distribution from its generating function. Example: Using telescope to measure intensity of an object
6.3 Stochastic Estimatio ad Cotrol, Fall 004 Lecture 7 Last time: Momets of the Poisso distributio from its geeratig fuctio. Gs () e dg µ e ds dg µ ( s) µ ( s) µ ( s) µ e ds dg X µ ds X s dg dg + ds ds
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 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 information7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7- Sectio 1. Samplig Distributio 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
More informationSolution. 1 Solutions of Homework 1. Sangchul Lee. October 27, Problem 1.1
Solutio Sagchul Lee October 7, 017 1 Solutios of Homework 1 Problem 1.1 Let Ω,F,P) be a probability space. Show that if {A : N} F such that A := lim A exists, the PA) = lim PA ). Proof. Usig the cotiuity
More informationLecture 19: Convergence
Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may
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 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 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 informationSimulation. Two Rule For Inverting A Distribution Function
Simulatio Two Rule For Ivertig A Distributio Fuctio Rule 1. If F(x) = u is costat o a iterval [x 1, x 2 ), the the uiform value u is mapped oto x 2 through the iversio process. Rule 2. If there is a jump
More informationModeling and Performance Analysis with Discrete-Event Simulation
Simulatio Modelig ad Performace Aalysis with Discrete-Evet Simulatio Chapter 5 Statistical Models i Simulatio Cotets Basic Probability Theory Cocepts Useful Statistical Models Discrete Distributios Cotiuous
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 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 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 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 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 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 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 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 informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationChapter 2 The Monte Carlo Method
Chapter 2 The Mote Carlo Method The Mote Carlo Method stads for a broad class of computatioal algorithms that rely o radom sampligs. It is ofte used i physical ad mathematical problems ad is most useful
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 informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
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 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 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 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 informationSTAT 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
More informationNOTES ON DISTRIBUTIONS
NOTES ON DISTRIBUTIONS MICHAEL N KATEHAKIS Radom Variables Radom variables represet outcomes from radom pheomea They are specified by two objects The rage R of possible values ad the frequecy fx with which
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 informationLecture Notes for Analysis Class
Lecture Notes for Aalysis Class Topological Spaces A topology for a set X is a collectio T of subsets of X such that: (a) X ad the empty set are i T (b) Uios of elemets of T are i T (c) Fiite itersectios
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 informationLecture 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
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 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 informationLet us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.
Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,
More informationMath 10A final exam, December 16, 2016
Please put away all books, calculators, cell phoes ad other devices. You may cosult a sigle two-sided sheet of otes. Please write carefully ad clearly, USING WORDS (ot just symbols). Remember that the
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 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 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 informationIf a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?
2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a
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 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 informationHypothesis Testing. Evaluation of Performance of Learned h. Issues. Trade-off Between Bias and Variance
Hypothesis Testig Empirically evaluatig accuracy of hypotheses: importat activity i ML. Three questios: Give observed accuracy over a sample set, how well does this estimate apply over additioal samples?
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 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 informationEE 4TM4: Digital Communications II Probability Theory
1 EE 4TM4: Digital Commuicatios II Probability Theory I. RANDOM VARIABLES A radom variable is a real-valued fuctio defied o the sample space. Example: Suppose that our experimet cosists of tossig two fair
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 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 information[ ] ( ) ( ) [ ] ( ) 1 [ ] [ ] Sums of Random Variables Y = a 1 X 1 + a 2 X 2 + +a n X n The expected value of Y is:
PROBABILITY FUNCTIONS A radom variable X has a probabilit associated with each of its possible values. The probabilit is termed a discrete probabilit if X ca assume ol discrete values, or X = x, x, x 3,,
More informationReview on Probability Distributions
Discrete Probability Distributios: Biomial Probability Distributio ourse: MAT, Istructor: Md. Saifuddi Khalid Review o Probability Distributios Radom Variable. A radom variable is a variable which takes
More informationPRACTICE FINAL/STUDY GUIDE SOLUTIONS
Last edited December 9, 03 at 4:33pm) Feel free to sed me ay feedback, icludig commets, typos, ad mathematical errors Problem Give the precise meaig of the followig statemets i) a f) L ii) a + f) L iii)
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 informationEcon 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1.
Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chi-square Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio
More informationLecture 3 The Lebesgue Integral
Lecture 3: The Lebesgue Itegral 1 of 14 Course: Theory of Probability I Term: Fall 2013 Istructor: Gorda Zitkovic Lecture 3 The Lebesgue Itegral The costructio of the itegral Uless expressly specified
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 informationDistribution of Random Samples & Limit theorems
STAT/MATH 395 A - PROBABILITY II UW Witer Quarter 2017 Néhémy Lim Distributio of Radom Samples & Limit theorems 1 Distributio of i.i.d. Samples Motivatig example. Assume that the goal of a study is to
More information( ) = p and P( i = b) = q.
MATH 540 Radom Walks Part 1 A radom walk X is special stochastic process that measures the height (or value) of a particle that radomly moves upward or dowward certai fixed amouts o each uit icremet of
More informationJoint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { }
UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig
More 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 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 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 information= p x (1 p) 1 x. Var (X) =p(1 p) M X (t) =1+p(e t 1).
Prob. fuctio:, =1 () = 1, =0 = (1 ) 1 E(X) = Var (X) =(1 ) M X (t) =1+(e t 1). 1.1.2 Biomial distributio Parameter: 0 1; >0; MGF: M X (t) ={1+(e t 1)}. Cosider a sequece of ideedet Ber() trials. If X =
More informationSeptember 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1
September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright
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 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 informationTopic 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
More informationStatistical Signal Processing
ELEG-66 Statistical Sigal Processig Pro. Barer 6 Evas Hall 8-697 barer@udel.edu Goal: Give a discrete time sequece {, how we develop Statistical ad spectral represetatios Filterig, predictio, ad sstem
More informationSolutions to Homework 2 - Probability Review
Solutios to Homework 2 - Probability Review Beroulli, biomial, Poisso ad ormal distributios. A Biomial distributio. Sice X is a biomial RV with parameters, p), it ca be writte as X = B i ) where B,...,
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 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 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 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 informationECE 330:541, Stochastic Signals and Systems Lecture Notes on Limit Theorems from Probability Fall 2002
ECE 330:541, Stochastic Sigals ad Systems Lecture Notes o Limit Theorems from robability Fall 00 I practice, there are two ways we ca costruct a ew sequece of radom variables from a old sequece of radom
More informationPRELIM PROBLEM SOLUTIONS
PRELIM PROBLEM SOLUTIONS THE GRAD STUDENTS + KEN Cotets. Complex Aalysis Practice Problems 2. 2. Real Aalysis Practice Problems 2. 4 3. Algebra Practice Problems 2. 8. Complex Aalysis Practice Problems
More informationLast Lecture. Wald Test
Last Lecture Biostatistics 602 - Statistical Iferece Lecture 22 Hyu Mi Kag April 9th, 2013 Is the exact distributio of LRT statistic typically easy to obtai? How about its asymptotic distributio? For testig
More informationCastiel, Supernatural, Season 6, Episode 18
13 Differetial Equatios the aswer to your questio ca best be epressed as a series of partial differetial equatios... Castiel, Superatural, Seaso 6, Episode 18 A differetial equatio is a mathematical equatio
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 informationModule 1 Fundamentals in statistics
Normal Distributio Repeated observatios that differ because of experimetal error ofte vary about some cetral value i a roughly symmetrical distributio i which small deviatios occur much more frequetly
More informationLimit Theorems. Convergence in Probability. Let X be the number of heads observed in n tosses. Then, E[X] = np and Var[X] = np(1-p).
Limit Theorems Covergece i Probability Let X be the umber of heads observed i tosses. The, E[X] = p ad Var[X] = p(-p). L O This P x p NM QP P x p should be close to uity for large if our ituitio is correct.
More information