Stochastic Processes
|
|
- Edgar Webb
- 5 years ago
- Views:
Transcription
1 Stochastic Processes Review o Elemetar Probabilit Lecture I Hamid R. Rabiee Fall 20 Ali Jalali
2 Outlie Histor/Philosoph Radom Variables Desit/Distributio Fuctios Joit/Coditioal Distributios Correlatio Importat Theorems
3 Histor & Philosoph Started b gamblers dispute Probabilit as a game aalzer! Formulated b B. Pascal ad P. Fermet First Problem (654) : Double Si durig 24 throws First Book (657) : Christia Huges De Ratiociiis i Ludo Aleae I Germa 657.
4 Histor & Philosoph (Cot d) Rapid developmet durig 8 th Cetur Major Cotributios: J. Beroulli ( ) A. De Moivre ( )
5 Histor & Philosoph (Cot d) A reaissace: Geeralizig the cocepts rom mathematical aalsis o games to aalzig scietiic ad practical problems: P. Laplace ( ) New approach irst book: P. Laplace Théorie Aaltique des Probabilités I Frace 82.
6 Histor & Philosoph (Cot d) 9 th cetur s developmets: Theor o errors Actuarial mathematics Statistical mechaics Other giats i the ield: Chebshev Markov ad Kolmogorov
7 Histor & Philosoph (Cot d) Moder theor o probabilit (20 th ) : A. Kolmogorov : Aiomatic approach First moder book: A. Kolmogorov Foudatios o Probabilit Theor Chelsea New ork 950 Nowadas Probabilit theor as a part o a theor called Measure theor!
8 Histor & Philosoph (Cot d) Two major philosophies: Frequetist Philosoph Observatio is eough Baesia Philosoph Observatio is NOT eough Prior kowledge is essetial Both are useul
9 Histor & Philosoph (Cot d) Frequetist philosoph There eist ied parameters like mea. There is a uderlig distributio rom which samples are draw Likelihood uctios(l()) maimize parameter/data For Gaussia distributio the L() or the mea happes to be /N i i or the average. Baesia philosoph Parameters are variable Variatio o the parameter deied b the prior probabilit This is combied with sample data p(/) to update the posterior distributio p(/). Mea o the posterior p(/)ca be cosidered a poit estimate o.
10 Histor & Philosoph (Cot d) A Eample: A coi is tossed 000 times ieldig 800 heads ad 200 tails. Let p = P(heads) be the bias o the coi. What is p? Baesia Aalsis Our prior kowledge (believe) : p (Uiorm(0)) Our posterior kowledge : pobservatio p 800 p Frequetist Aalsis Aswer is a estimator pˆ such that 8 Mea : E pˆ 0. Coidece Iterval : P pˆ
11 Histor & Philosoph (Cot d) Further readig: hist/stathist.htm stat/histor/idehistor.shtml istprob.pd
12 Outlie Histor/Philosoph Radom Variables Desit/Distributio Fuctios Joit/Coditioal Distributios Correlatio Importat Theorems
13 Radom Variables F P F F P P
14 Radom Variables (Cot d) Radom variable is a uctio ( mappig ) rom a set o possible outcomes o the eperimet to a iterval o real (comple) umbers. I other words : Outcomes F I R : : F r I Real Lie
15 Radom Variables (Cot d) Eample I : Mappig aces o a dice to the irst si atural umbers. Eample II : Mappig height o a ma to the real iterval (03] (meter or somethig else). Eample III : Mappig success i a eam to the discrete iterval [020] b quatum 0..
16 Radom Variables (Cot d) Radom Variables Discrete Dice Coi Grade o a course etc. Cotiuous Temperature Humidit Legth etc. Radom Variables Real Comple
17 Outlie Histor/Philosoph Radom Variables Desit/Distributio Fuctios Joit/Coditioal Distributios Correlatio Importat Theorems
18 Desit/Distributio Fuctios Probabilit Mass Fuctio (PMF) Discrete radom variables Summatio o impulses The magitude o each impulse represets the probabilit o occurrece o the outcome P Eample I: Rollig a air dice PMF 6 6 i i
19 Desit/Distributio Fuctios (Cot d) Eample II: Summatio o two air dices P Note : Summatio o all probabilities should be equal to ONE. (Wh?)
20 Desit/Distributio Fuctios (Cot d) Probabilit Desit Fuctio (PDF) Cotiuous radom variables The probabilit o occurrece o will be P d. 0 d 2 d 2 P P
21 Desit/Distributio Fuctios (Cot d) Some amous masses ad desities Uiorm Desit P a. U ed Ubegi a Gaussia (Normal) Desit. 2 P a. e 2 2. N 2 2
22 Desit/Distributio Fuctios (Cot d) Biomial Desit Poisso Desit! : Note e p N p N.. 0 N.p N Importat Fact:!... :. p N e p p N llarge N For Suiciet p N N
23 Desit/Distributio Fuctios (Cot d) Cauch Desit P 2 2 Weibull Desit k k e k
24 Desit/Distributio Fuctios (Cot d) Epoetial Desit Raleigh Desit e U e e
25 Desit/Distributio Fuctios (Cot d) Epected Value The most likelihood value Liear Operator Fuctio o a radom variable Epectatio E E E. d a. b a. E b g g. d
26 Desit/Distributio Fuctios (Cot d) PDF o a uctio o radom variables Assume RV such that g The iverse equatio g ma have more tha oe solutio called... 2 PDF o ca be obtaied rom PDF o as ollows i d d g i i
27 Desit/Distributio Fuctios (Cot d) Cumulative Distributio Fuctio (CDF) Both Cotiuous ad Discrete Could be deied as the itegratio o PDF PDF CDF F F P. d CDF()
28 Desit/Distributio Fuctios (Cot d) Some CDF properties No-decreasig Right Cotiuous F(-iiit) = 0 F(iiit) =
29 Outlie Histor/Philosoph Radom Variables Desit/Distributio Fuctios Joit/Coditioal Distributios Correlatio Importat Theorems
30 Joit/Coditioal Distributios Joit Probabilit Fuctios Desit Distributio Eample I I a rollig air dice eperimet represet the outcome as a 3-bit digital umber z... 0 ; 6 0 ; 3 0; 3 0 0; 6 O W z dd ad P F
31 Joit/Coditioal Distributios (Cot d) Eample II Two ormal radom variables What is r? Idepedet Evets (Strog Aiom) r r e r
32 Joit/Coditioal Distributios (Cot d) Obtaiig oe variable desit uctios Distributio uctios ca be obtaied just rom the desit uctios. (How?) d d
33 Joit/Coditioal Distributios (Cot d) Coditioal Desit Fuctio Probabilit o occurrece o a evet i aother evet is observed (we kow what is). Baes Rule d..
34 Joit/Coditioal Distributios (Cot d) Eample I Rollig a air dice : the outcome is a eve umber : the outcome is a prime umber Eample II Joit ormal (Gaussia) radom variables P P P r r e r
35 Joit/Coditioal Distributios (Cot d) Coditioal Distributio Fuctio Note that is a costat durig the itegratio. dt t dt t d while P F
36 Joit/Coditioal Distributios (Cot d) Idepedet Radom Variables Remember! Idepedec is NOT heuristic..
37 Joit/Coditioal Distributios (Cot d) PDF o a uctios o joit radom variables Assume that The iverse equatio set has a set o solutios Deie Jacobea matri as ollows The joit PDF will be g V U ) ( U V g ) ( V U V U J i i i U V i i J determiat absolute v u.
38 Outlie Histor/Philosoph Radom Variables Desit/Distributio Fuctios Joit/Coditioal Distributios Correlatio Importat Theorems
39 Correlatio Kowig about a radom variable how much iormatio will we gai about the other radom variable? Shows liear similarit More ormal: E Crr. Covariace is ormalized correlatio. E.. Cov( ) E
40 Correlatio (cot d) Variace Covariace o a radom variable with itsel Var 2 E Relatio betwee correlatio ad covariace Stadard Deviatio Square root o variace E
41 Correlatio (cot d) Momets th order momet o a radom variable is the epected value o M E Normalized orm M E Mea is irst momet Variace is secod momet added b the square o the mea
42 Outlie Histor/Philosoph Radom Variables Desit/Distributio Fuctios Joit/Coditioal Distributios Correlatio Importat Theorems
43 Importat Theorems Cetral limit theorem Suppose i.i.d. (Idepedet Ideticall Distributed) RVs k with iite variaces Let S i a. PDF o S coverges to a ormal distributio as icreases regardless to the desit o RVs. Eceptio : Cauch Distributio (Wh?)
44 Importat Theorems (cot d) Law o Large Numbers (Weak) For i.i.d. RVs k 0 Pr lim 0 i i
45 Importat Theorems (cot d) Law o Large Numbers (Strog) For i.i.d. RVs k Wh this deiitio is stroger tha beore? lim Pr i i
46 Importat Theorems (cot d) Chebshev s Iequalit Let be a oegative RV Let c be a positive umber Aother orm: Pr Pr c c E 2 2 It could be rewritte or egative RVs. (How?)
47 Importat Theorems (cot d) Schwarz Iequalit For two RVs ad with iite secod momets E E.E Equalit holds i case o liear depedec.
48 Net Lecture Elemets o Stochastic Processes
Review of Elementary Probability Lecture I Hamid R. Rabiee
Stochastic Processes Review o Elementar Probabilit Lecture I Hamid R. Rabiee Outline Histor/Philosoph Random Variables Densit/Distribution Functions Joint/Conditional Distributions Correlation Important
More informationStochastic Processes. Review of Elementary Probability Lecture I. Hamid R. Rabiee Ali Jalali
Stochastic Processes Review o Elementary Probability bili Lecture I Hamid R. Rabiee Ali Jalali Outline History/Philosophy Random Variables Density/Distribution Functions Joint/Conditional Distributions
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 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 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 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 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 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 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 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 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 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 informationNotation List. For Cambridge International Mathematics Qualifications. For use from 2020
Notatio List For Cambridge Iteratioal Mathematics Qualificatios For use from 2020 Notatio List for Cambridge Iteratioal Mathematics Qualificatios (For use from 2020) Mathematical otatio Eamiatios for CIE
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 informationSTAT 516 Answers Homework 6 April 2, 2008 Solutions by Mark Daniel Ward PROBLEMS
STAT 56 Aswers Homework 6 April 2, 28 Solutios by Mark Daiel Ward PROBLEMS Chapter 6 Problems 2a. The mass p(, correspods to either o the irst two balls beig white, so p(, 8 7 4/39. The mass p(, correspods
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 informationAMS 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
More informationMATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4
MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.
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 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 informationContinuous Random Variables: Conditioning, Expectation and Independence
Cotiuous Radom Variables: Coditioig, Expectatio ad Idepedece Berli Che Departmet o Computer ciece & Iormatio Egieerig Natioal Taiwa Normal Uiversit Reerece: - D.. Bertsekas, J. N. Tsitsiklis, Itroductio
More informationThis section is optional.
4 Momet Geeratig Fuctios* This sectio is optioal. The momet geeratig fuctio g : R R of a radom variable X is defied as g(t) = E[e tx ]. Propositio 1. We have g () (0) = E[X ] for = 1, 2,... Proof. Therefore
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 informationProbability review (week 2) Solutions
Probability review (week 2) Solutios A. Biomial distributio. BERNOULLI, BINOMIAL, POISSON AND NORMAL DISTRIBUTIONS. X is a biomial RV with parameters,p. Let u i be a Beroulli RV with probability of success
More informationDifferential Entropy
School o Iormatio Sciece Dieretial Etropy 009 - Course - Iormatio Theory - Tetsuo Asao ad Tad matsumoto Email: {t-asao matumoto}@jaist.ac.jp Japa Advaced Istitute o Sciece ad Techology Asahidai - Nomi
More informationProbability and MLE.
10-701 Probability ad MLE http://www.cs.cmu.edu/~pradeepr/701 (brief) itro to probability Basic otatios Radom variable - referrig to a elemet / evet whose status is ukow: A = it will rai tomorrow Domai
More informationNANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS
NANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER. There will be oe 2-hour paper cosistig of 4 questios.
More informationHOMEWORK I: PREREQUISITES FROM MATH 727
HOMEWORK I: PREREQUISITES FROM MATH 727 Questio. Let X, X 2,... be idepedet expoetial radom variables with mea µ. (a) Show that for Z +, we have EX µ!. (b) Show that almost surely, X + + X (c) Fid the
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 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 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 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 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 informationCS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 5
CS434a/54a: Patter Recogitio Prof. Olga Veksler Lecture 5 Today Itroductio to parameter estimatio Two methods for parameter estimatio Maimum Likelihood Estimatio Bayesia Estimatio Itroducto Bayesia Decisio
More informationLecture 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
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 Poisson Process *
OpeStax-CNX module: m11255 1 The Poisso Process * Do Johso This work is produced by OpeStax-CNX ad licesed uder the Creative Commos Attributio Licese 1.0 Some sigals have o waveform. Cosider the measuremet
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationAdvanced Stochastic Processes.
Advaced Stochastic Processes. David Gamarik LECTURE 2 Radom variables ad measurable fuctios. Strog Law of Large Numbers (SLLN). Scary stuff cotiued... Outlie of Lecture Radom variables ad measurable fuctios.
More 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 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 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 informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
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 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 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 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 information7.1 Convergence of sequences of random variables
Chapter 7 Limit theorems Throughout this sectio we will assume a probability space (Ω, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More 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 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 informationSTATISTICAL METHODS FOR BUSINESS
STATISTICAL METHODS FOR BUSINESS UNIT 5. Joit aalysis ad limit theorems. 5.1.- -dimesio distributios. Margial ad coditioal distributios 5.2.- Sequeces of idepedet radom variables. Properties 5.3.- Sums
More informationMathematical Statistics - MS
Paper Specific Istructios. The examiatio is of hours duratio. There are a total of 60 questios carryig 00 marks. The etire paper is divided ito three sectios, A, B ad C. All sectios are compulsory. Questios
More informationAdvanced Engineering Mathematics Exercises on Module 4: Probability and Statistics
Advaced Egieerig Mathematics Eercises o Module 4: Probability ad Statistics. A survey of people i give regio showed that 5% drak regularly. The probability of death due to liver disease, give that a perso
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 informationStatistics Fall 2004 Theory of Probability Practice Final # 1 { Solutions
Statistics 6 - Fall 4 Theor of Probabilit Practice Fial # { Solutios Istructios Aswer Q. -6. All questios have equal weight. Q. Let X ad Y be idepedet, both with Geometric distributio Geom(p For itegers
More informationFundamental Concepts: Surfaces and Curves
UNDAMENTAL CONCEPTS: SURACES AND CURVES CHAPTER udametal Cocepts: Surfaces ad Curves. INTRODUCTION This chapter describes two geometrical objects, vi., surfaces ad curves because the pla a ver importat
More informationIIT JAM Mathematical Statistics (MS) 2006 SECTION A
IIT JAM Mathematical Statistics (MS) 6 SECTION A. If a > for ad lim a / L >, the which of the followig series is ot coverget? (a) (b) (c) (d) (d) = = a = a = a a + / a lim a a / + = lim a / a / + = lim
More informationCS537. Numerical Analysis and Computing
CS57 Numerical Aalysis ad Computig Lecture Locatig Roots o Equatios Proessor Ju Zhag Departmet o Computer Sciece Uiversity o Ketucky Leigto KY 456-6 Jauary 9 9 What is the Root May physical system ca be
More informationChapter 22. Comparing Two Proportions. Copyright 2010, 2007, 2004 Pearson Education, Inc.
Chapter 22 Comparig Two Proportios Copyright 2010, 2007, 2004 Pearso Educatio, Ic. Comparig Two Proportios Read the first two paragraphs of pg 504. Comparisos betwee two percetages are much more commo
More information(6) Fundamental Sampling Distribution and Data Discription
34 Stat Lecture Notes (6) Fudametal Samplig Distributio ad Data Discriptio ( Book*: Chapter 8,pg5) Probability& Statistics for Egieers & Scietists By Walpole, Myers, Myers, Ye 8.1 Radom Samplig: Populatio:
More informationf(x i ; ) L(x; p) = i=1 To estimate the value of that maximizes L or equivalently ln L we will set =0, for i =1, 2,...,m p x i (1 p) 1 x i i=1
Parameter Estimatio Samples from a probability distributio F () are: [,,..., ] T.Theprobabilitydistributio has a parameter vector [,,..., m ] T. Estimator: Statistic used to estimate ukow. Estimate: Observed
More informationSampling Error. Chapter 6 Student Lecture Notes 6-1. Business Statistics: A Decision-Making Approach, 6e. Chapter Goals
Chapter 6 Studet Lecture Notes 6-1 Busiess Statistics: A Decisio-Makig Approach 6 th Editio Chapter 6 Itroductio to Samplig Distributios Chap 6-1 Chapter Goals After completig this chapter, you should
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 informationJanuary 25, 2017 INTRODUCTION TO MATHEMATICAL STATISTICS
Jauary 25, 207 INTRODUCTION TO MATHEMATICAL STATISTICS Abstract. A basic itroductio to statistics assumig kowledge of probability theory.. Probability I a typical udergraduate problem i probability, we
More information32 estimating the cumulative distribution function
32 estimatig the cumulative distributio fuctio 4.6 types of cofidece itervals/bads Let F be a class of distributio fuctios F ad let θ be some quatity of iterest, such as the mea of F or the whole fuctio
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 informationTopic 9 - Taylor and MacLaurin Series
Topic 9 - Taylor ad MacLauri Series A. Taylors Theorem. The use o power series is very commo i uctioal aalysis i act may useul ad commoly used uctios ca be writte as a power series ad this remarkable result
More informationA PROBABILITY PRIMER
CARLETON COLLEGE A ROBABILITY RIMER SCOTT BIERMAN (Do ot quote without permissio) A robability rimer INTRODUCTION The field of probability ad statistics provides a orgaizig framework for systematically
More informationThis exam contains 19 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam.
Probability ad Statistics FS 07 Secod Sessio Exam 09.0.08 Time Limit: 80 Miutes Name: Studet ID: This exam cotais 9 pages (icludig this cover page) ad 0 questios. A Formulae sheet is provided with the
More informationECONOMETRIC THEORY. MODULE XIII Lecture - 34 Asymptotic Theory and Stochastic Regressors
ECONOMETRIC THEORY MODULE XIII Lecture - 34 Asymptotic Theory ad Stochastic Regressors Dr. Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Asymptotic theory The asymptotic
More informationSampling Distributions, Z-Tests, Power
Samplig Distributios, Z-Tests, Power We draw ifereces about populatio parameters from sample statistics Sample proportio approximates populatio proportio Sample mea approximates populatio mea Sample variace
More informationThe Central Limit Theorem
Chapter The Cetral Limit Theorem Deote by Z the stadard ormal radom variable with desity 2π e x2 /2. Lemma.. Ee itz = e t2 /2 Proof. We use the same calculatio as for the momet geeratig fuctio: exp(itx
More informationElementary manipulations of probabilities
Elemetary maipulatios of probabilities Set probability of multi-valued r.v. {=Odd} = +3+5 = /6+/6+/6 = ½ X X,, X i j X i j Multi-variat distributio: Joit probability: X true true X X,, X X i j i j X X
More informationEcon 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
More informationz is the upper tail critical value from the normal distribution
Statistical Iferece drawig coclusios about a populatio parameter, based o a sample estimate. Populatio: GRE results for a ew eam format o the quatitative sectio Sample: =30 test scores Populatio Samplig
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 informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More information15-780: Graduate Artificial Intelligence. Density estimation
5-780: Graduate Artificial Itelligece Desity estimatio Coditioal Probability Tables (CPT) But where do we get them? P(B)=.05 B P(E)=. E P(A B,E) )=.95 P(A B, E) =.85 P(A B,E) )=.5 P(A B, E) =.05 A P(J
More informationMATH CALCULUS II Objectives and Notes for Test 4
MATH 44 - CALCULUS II Objectives ad Notes for Test 4 To do well o this test, ou should be able to work the followig tpes of problems. Fid a power series represetatio for a fuctio ad determie the radius
More informationINF Introduction to classifiction Anne Solberg Based on Chapter 2 ( ) in Duda and Hart: Pattern Classification
INF 4300 90 Itroductio to classifictio Ae Solberg ae@ifiuioo Based o Chapter -6 i Duda ad Hart: atter Classificatio 90 INF 4300 Madator proect Mai task: classificatio You must implemet a classificatio
More information2. The volume of the solid of revolution generated by revolving the area bounded by the
IIT JAM Mathematical Statistics (MS) Solved Paper. A eigevector of the matrix M= ( ) is (a) ( ) (b) ( ) (c) ( ) (d) ( ) Solutio: (a) Eigevalue of M = ( ) is. x So, let x = ( y) be the eigevector. z (M
More informationLecture 20: Multivariate convergence and the Central Limit Theorem
Lecture 20: Multivariate covergece ad the Cetral Limit Theorem Covergece i distributio for radom vectors Let Z,Z 1,Z 2,... be radom vectors o R k. If the cdf of Z is cotiuous, the we ca defie covergece
More informationIntroduction to probability Stochastic Process Queuing systems. TELE4642: Week2
Itroductio to probability Stochastic Process Queuig systems TELE4642: Week2 Overview Refresher: Probability theory Termiology, defiitio Coditioal probability, idepedece Radom variables ad distributios
More informationSTAT 515 fa 2016 Lec Sampling distribution of the mean, part 2 (central limit theorem)
STAT 515 fa 2016 Lec 15-16 Samplig distributio of the mea, part 2 cetral limit theorem Karl B. Gregory Moday, Sep 26th Cotets 1 The cetral limit theorem 1 1.1 The most importat theorem i statistics.............
More informationChapter 2 Transformations and Expectations
Chapter Trasformatios a Epectatios Chapter Distributios of Fuctios of a Raom Variable Problem: Let be a raom variable with cf F ( ) If we efie ay fuctio of, say g( ) g( ) is also a raom variable whose
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 informationLecture 2: Monte Carlo Simulation
STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: Ye-Chi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?
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 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 informationCommunications II Lecture 2: Probability and Random Processes
Commuicatio II Lecture : robabilit ad Radom rocee roeor Ki K. Leug ad Computig Departmet Imperial College Lodo Copright reerved Sigiicace o probabilit theor robabilit i the core mathematical tool or commuicatio
More informationPROBABILITY, STATISTICS, AND RANDOM PROCESSES EE 351K
PROBABILITY, STATISTICS, AND RANDOM PROCESSES EE 35K! factorial... 3, 4 st fudametal theorem of probability... 6 d fudametal theorem of probability... 7 absorbig matrices... absorptio probability... time...
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 2: Poisson Sta*s*cs Probability Density Func*ons Expecta*on and Variance Es*mators
Lecture 2: Poisso Sta*s*cs Probability Desity Fuc*os Expecta*o ad Variace Es*mators Biomial Distribu*o: P (k successes i attempts) =! k!( k)! p k s( p s ) k prob of each success Poisso Distributio Note
More informationLearning Theory: Lecture Notes
Learig Theory: Lecture Notes Kamalika Chaudhuri October 4, 0 Cocetratio of Averages Cocetratio of measure is very useful i showig bouds o the errors of machie-learig algorithms. We will begi with a basic
More informationCS321. Numerical Analysis and Computing
CS Numerical Aalysis ad Computig Lecture Locatig Roots o Equatios Proessor Ju Zhag Departmet o Computer Sciece Uiversity o Ketucky Leigto KY 456-6 September 8 5 What is the Root May physical system ca
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 informationExercise 4.3 Use the Continuity Theorem to prove the Cramér-Wold Theorem, Theorem. (1) φ a X(1).
Assigmet 7 Exercise 4.3 Use the Cotiuity Theorem to prove the Cramér-Wold Theorem, Theorem 4.12. Hit: a X d a X implies that φ a X (1) φ a X(1). Sketch of solutio: As we poited out i class, the oly tricky
More informationSTA 4032 Final Exam Formula Sheet
Chapter 2. Probability STA 4032 Fial Eam Formula Sheet Some Baic Probability Formula: (1) P (A B) = P (A) + P (B) P (A B). (2) P (A ) = 1 P (A) ( A i the complemet of A). (3) If S i a fiite ample pace
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 information