# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

2 Pages

### Math556-Ex5

Course: MATH 556, Fall 2009
School: McGill
Rating:

Word Count: 512

#### Document Preview

556 MATH - EXERCISES 5 These exercises are not for assessment 1. Using the Central Limit Theorem, construct Normal approximations to probability distribution of a random variable X having (i) (ii) (iii) (iv) a Binomial distribution, X Binomial(n, ) a Poisson distribution, X P oisson() a Negative Binomial distribution, X N egBinomial(n, ) a Gamma distribution,X Gamma(, ) In the following questions, use the...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Canada >> McGill >> MATH 556

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
556 MATH - EXERCISES 5 These exercises are not for assessment 1. Using the Central Limit Theorem, construct Normal approximations to probability distribution of a random variable X having (i) (ii) (iii) (iv) a Binomial distribution, X Binomial(n, ) a Poisson distribution, X P oisson() a Negative Binomial distribution, X N egBinomial(n, ) a Gamma distribution,X Gamma(, ) In the following questions, use the following results concerning extreme order statistics; let Yn and Zn correspond to the maximum and minimum order statistics derived from random sample X1 , ...Xn from population with cdf FX , that is Yn = max {X1 , ..., Xn } Then the cdfs of Yn and Zn are given by FYn (y) = {FX (y)}n 2. Suppose X1 , ..., Xn U nif orm(0, 1), that is FX (x) = x 0x1 FZn (z) = 1 {1 FX (z)}n . Zn = min {X1 , ..., Xn } . Find the cdfs of Yn and Zn , and the limiting distributions as n . 3. Suppose X1 , ..., Xn have cdf FX (x) = 1 x1 x1 n Find the cdfs of Zn and Un = Zn , and the limiting distributions of Zn and Un as n . 4. Suppose X1 , ..., Xn have cdf 1 xR 1 + ex Find the cdfs of Yn and Un = Yn log n and the limiting distributions of Yn and Un as n . FX (x) = 1 x>0 1 + x Find the cdfs of Yn and Zn , and the limiting distributions as n . Find also the cdfs of Un = Yn /n and Vn = nZn , and the limiting distributions of Un and Vn as n . FX (x) = 1 Suppose X1 , ..., Xn P oisson() are independent random variables. Let 1 Mn = n p n 5. Suppose X1 , ..., Xn have cdf 6. Xi i=1 Show Mn that as n . If random variable Tn is dened by Tn = eMn , show that p Tn e , and nd an approximation to the probability distribution of Tn as n . MATH 556 EXERCISES 5 Page 1 of 2 7. For the following sequences of random variables, {Xn }, decide whether the the sequence converges in mean-square (rth mean for r = 2) or in probability as n . 1 with prob. 1/n (a) Xn = 2 with prob. 1 1/n n2 with prob. 1/n (b) Xn = 1 with prob. 1 1/n n with prob. 1/ log n (c) Xn = 0 with prob. 1 1/ log n Almost sure convergence and the Borel-Cantelli Lemma. 8. Consider the sequence of random variables dened for n = 1, 2, 3, ... by Xn = I[0,n1 ) (Un ) where U1 , U2 , ... are a sequence of independent U nif orm(0, 1) random variables, and IA is the indicator function for set A 1 A IA () = 0 A / Does the sequence {Xn } converge (a) (b) almost surely ? in rth mean for r = 1 ? [Hint: ...

Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Sveriges lantbruksuniversitet - ENSC - 428
Simon Fraser University School of Engineering Science ENSC 428-4 Data Communications Spring 2001 Calendar Description This course will cover the physical-layer design issues in digital communication systems. The major topics covered are: information
McGill - MATH - 556
556: M ATHEMATICAL S TATISTICS IT HE K ULLBACK -L EIBLER D IVERGENCE The Kullback-Leibler (KL) divergence between two probability distributions with densities f0 and f1 with supports X0 and X1 respectively is defined asK (f0 , f1 ) =logX0f0 (X
McGill - MATH - 556
556: M ATHEMATICAL S TATISTICS IS TOCHASTIC C ONVERGENCE7Convergence ConceptsThe following denitions are stated in terms of scalar random variables, but extend naturally to vector random variables dened on the same probability space with measu
Sveriges lantbruksuniversitet - ENSC - 805
TRANSACTIONS IEEECOMMUNICATIONS, ON COM-32, VOL.NO. 10, OCTOBER 198410791983.[6] L. G . MasonandT.Gavin,An application of optimal control theory [9] B. A.Murtaughand M. A. Saunders, Large-scale lineariy conto modernization problems in telep
Sveriges lantbruksuniversitet - ENSC - 805
1744IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 9, SEPTEMBER 2001Turbo Equalization: Adaptive Equalization and Channel Decoding Jointly OptimizedChristophe Laot, Alain Glavieux, and Jol LabatAbstract-This paper deals with a r
Sveriges lantbruksuniversitet - BUS - 362
BUS 362: The Anal ysi s and Desi gn of Busi ness I nfor mati on Systemshttp:/ / mi s.bus.sfu.ca/ bus362Drew Parkerdr ew@ sfu.cahttp:/ / par ker .bus.sfu.caWeek 4 Agenda Whats new in technology this week? A couple of Interesting Questions to
Wilfrid Laurier - CPSC - 441
Link-State and Distance Vector Routing ExamplesCPSC 441 University of CalgaryLink-State (LS) Routing AlgorithmDijkstra's algorithm topology and link costs known to all nodes accomplished via &quot;link state broadcast&quot; all nodes have same info compute
Wilfrid Laurier - CPSC - 441
Assignment 3Shortest Path and Dijkstras Algorithm On a network, there may be more than paths from a certain node to another one The path with the smallest weight sum is called the shortest path between the two nodes. Dijkstras algorithm offers a
East Los Angeles College - PHY - 112
PHY112 Semester 1IntegrationDr. Tim RichardsonContentsLecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 Lecture 8 Indefinite integrals and signed area Definite integrals Improper, complex and symmetric integrals Methods of i
McGill - MATH - 556
556: M ATHEMATICAL S TATISTICS I C OMPUTING THE H YPERBOLIC S ECANT D ISTRIBUTION C HARACTERISTIC F UNCTIONDavid A. Stephens Department of Mathematics and Statistics McGill University October 28, 2006Abstract We give two methods for computing the c
East Los Angeles College - PHY - 112
Vectors Lecture 1INTRODUCTION: SCALAR AND VECTOR QUANTITIES what is a vector? We can divide physical quantities into two main categories: scalar and vector quantities. A scalar quantity is described completely by a single value or number with appro
East Los Angeles College - PHY - 112
LECTURES 4 &amp; 5 METHODS OF INTEGRATIONA Substitution B Logarithmic integration (very brief) C Inspection (very brief) A Substitution This is probably the most common method used to carry out integration. We'll look at various kinds of functions:
Wilfrid Laurier - CPSC - 587
Inverse KinematicsInput:The desired position and possibly orientation of the end effectorOutput:The state of the component joints.
East Los Angeles College - PHY - 112
PHY112 Introductory mathematics for physicists and astronomers 2008/2009ScheduleUnit 1 Introduction 2 Functions and differentiation 3 Complex numbers 4 Integration 5 Differential equations 6 Vectors 7 Functions of two variables 8 Probability Lectu
McGill - MATH - 556
MATH 556 - ASSIGNMENT 4To be handed in not later than 5pm, 27th November 2008. Please hand in during lectures, to Burnside 1235, or to the Mathematics Office Burnside 1005 1 (a) Let r &gt; s 1. Prove that, for convergence in mean, Xn - Xrth=Xn -
East Los Angeles College - PHY - 112
PHY112 Maths for Physicists and AstronomersTutorial Sheet 1 Basic Algebra Revision For the PHY112 Maths tutorial classes in Week 2 (week beginning Monday 6th October).Homework submit your answers to your tutor during the class.1 Expand (remove t
East Los Angeles College - PHY - 112
PHY112 Unit 1: Introduction D J MowbrayPHY112 Introductory mathematics for physicists and astronomersUnit 1: IntroductionPartial fractions and the manipulation of power indicies Partial fractions You will often meet functions of the form 2 x ( x
Allan Hancock College - MINE - 2106
MINE2106Method by Sections Project ReportPractical 4Day Dawn Prospect Method by Sections This practical and associated assignment is based on a gold prospect in north Queensland that was first worked in the 1890's when a shaft and several pits
East Los Angeles College - PHY - 112
Vectors Lecture 2RELATIVE VELOCITY Usually, adding velocity vectors is concerned with combining relative velocities. If an escalator is moving upwards at 1.0 ms-1 relative to the ground floor and a person stood on it is walking up at 1.1 ms-1, then
East Los Angeles College - PHY - 112
PHY112 Unit 3: Complex numbers D J MowbrayPHY112 Introductory mathematics for physicists and astronomersUnit 3: Complex numbersContents 1. Definition of a complex number 2. The Argand diagram 3. Modulus, argument and conjugate of a complex numbe
East Los Angeles College - PHY - 112
PHY112 - Unit 5: Differential equationsV. A. Kudryavtsev Autumn 2008Web-page: http:/kudryavtsev.staff.shef.ac.uk/phy112/ or link from 'list of modules'1Introduction, examplesThe differential equation is an equation which includes one (or mor
St. Mary MD - CS - 4360
Allan Hancock College - ERTH - 3502
ERTH3502Magmatic Ore-forming ProcessesLecture 8Magmatic ore-forming processes:Crystal-liquid fractionation mafic and ultramafic magmas early formed crystals form cumulate layers in magma chamber magma mixing required to explain oxide only c
Sveriges lantbruksuniversitet - CS - 710
Solutions to the second assignment, CMPT 710 Posted: February 23, 2009 1. Let f and g be two functions computable in logarithmic space. Prove that their composition, the function x f (g(x) is also logspace computable. The trick is not to compute g(x
East Los Angeles College - ZOOL - 0601
BMC BiologyResearch articleBioMed CentralOpen AccessCooperation and virulence in acute Pseudomonas aeruginosa infectionsFreya Harrison*1, Lucy E Browning1,2, Michiel Vos3 and Angus Buckling1Address: 1Department of Zoology, University of Oxfo
East Los Angeles College - MAGD - 1129
Reflections on externalism and self-knowledgeIAN PHILLIPS, MindGrad Meeting 13/6/061. Introduction In the mid-nineties a large number of philosophers (most famously, Michael McKinsey, Jessica Brown and Paul Boghossian) raised and discussed a certa
East Los Angeles College - SHUG - 2184
Scallop heavy meromyosin is regulated by calciumDavid Yadin, St. Hugh's College Oxford Work carried out at Division of Physical Biochemistry, NIMR, Mill HillNull Hypothe incre sis: asing thecalciumion conce ntration has no e ct on theactivity of s
East Los Angeles College - BALL - 0888
Work for Week 3Part [A]: Reprise on ValidityTwo extracts from past papers. Exam standards, please. Which you now know to mean Brixel standards. Q.1. Can there be valid arguments where (i) (ii) (iii) (iv) (v) the premisses are all false and so is th
Sveriges lantbruksuniversitet - CS - 710
Second assignment, CMPT 710 Posted: February 5, 2009 Due: February 18, 2009, 11:30am (beginning of class) Try to solve the problems yourself and resist the temptation to look up a solution or to discuss it with other students. It's better to ask me f
Sveriges lantbruksuniversitet - CS - 404
Probability Introduction ReminderCryptography and Protocols Andrei BulatovCryptography and Protocols - Probability3-2Sample Space and OutcomesExperiment s and outcomes Sample space is the set of all possible outcomes Examples - flipping a co
Toledo - BIO - 464
We can use allele frequencies (number of alleles and distributions of alleles over geographical space) to infer whether a species exists in one or many populations. Under the Island Model of Migration, a large population is split into subpopulations
East Los Angeles College - STUDIO - 0506
East Los Angeles College - STUDIO - 0506
project programphase 1ministry of state community consultation groupsan outline of the process of building the scheme, and when other actors will influence its processfull time employeeworkshop memberproject initiation funded by government:
East Los Angeles College - STUDIO - 0506
What is undeniable is that the landmark Sandy Row bonfire has been progressively squeezed. For many years it was on the site of Cliff-face flats, then, when that site was lost, on waste ground at the corner of Hope Street.[1] Sandy RowApproximate
Sveriges lantbruksuniversitet - CS - 404
Cryptography and Protocols Public Key Cryptography II14-2Modular ArithmeticWe define addition, subtraction, and multiplication of residues: Let a,b Z n . Then a + b (mod n) is the element c Z n such that c a + b (mod m) a b (mod n) is the e
East Los Angeles College - STUDIO - 0506
Local ResidenceAnnual Crop YieldAmount of food needed to sustain the population per annum (national food survey 2000)Area of Land AvailableTotal Available Land Used for Scheme 35600m2 0.036km2Population that can benefit from the schemeEmpty
Sveriges lantbruksuniversitet - CS - 404
Passwords IntroductionCryptography and Protocols Andrei BulatovCryptography and Protocols Passwords23-2Zero-Knowledge ProofsThe idea is to prove that User knows the password without sending it We use Diffie-Hellman key exchange protocol Fix
East Los Angeles College - STUDIO - 0506
PROGRAMURBAN PROTOTYPEAN Y E AC PEREQUIREMENTS/CONDITIONSAD RO E AC SP ID VO SS CE S) AC IDE EN S OP WO SS (T CE AC ) ED E IT SID M LI NE (O RY RA RE PO U M CT TE RU ST T EN E AN UR RM C T D PE R U NE ST OW Y L EL AL AT W IV E PR EAC L P AL W G
Sveriges lantbruksuniversitet - CS - 404
Pseudorandom Permutations Introduction Ciphers and BlockCryptography and Protocols Andrei BulatovCryptography and Protocols Block Ciphers9-2Pseudorandom PermutationsA pseudorandom function F = { f s }s{0,1}* is called a pseudorandom permuta
East Los Angeles College - STUDIO - 0506
Sveriges lantbruksuniversitet - CS - 404
East Los Angeles College - STUDIO - 0506
Sveriges lantbruksuniversitet - CS - 404
Cryptography and Protocols Perfect Security5-2Symmetric Encryption SchemeA symmetric encryption scheme is a triple of algorithms (K,E,D) - K keys generation - E encryption algorithm - D decryption algorithm For simplicity assume that k K unifo
East Los Angeles College - STUDIO - 0506
recyclable material processescollectionrequired staff mobile processing equipmentconveyor beltprocessing of materials analysed. equipment icons at 1:200 scalesortingbalingstacksdistributionnotesaround 11 people will be required to mai
Sveriges lantbruksuniversitet - CS - 404
Cryptography and Protocols Passwords23-2Zero-Knowledge ProofsThe idea is to prove that User knows the password without sending it We use Diffie-Hellman key exchange protocol Fix prime p and a primitive root g modulo pgXPasswords Introduction
East Los Angeles College - STUDIO - 0506
CONDITION 1CErase Notice Transform Move E: (E) (N) (T) (M) Catholic Resident (negative to changes) Catholic Resident (focused and positive) Children (N/A) ANGEL Police (passionate)CONDITION 2CErase Notice Transform Move E: N: T: (E) (N) (T) (M) I
Sveriges lantbruksuniversitet - CS - 404
Cryptography and Protocols - Classical Cryptosystems2-2NotationmessageClassical Cryptosystems IntroductionAlice Eve Plaintext Ciphertext Key Protocol: (K, E, D) K key generation algorithm E encryption algorithm D decryption algorithmBo
Sveriges lantbruksuniversitet - CS - 404
Classical Cryptosystems IntroductionCryptography and Protocols Andrei BulatovCryptography and Protocols - Classical Cryptosystems2-2Notationmessage Alice Eve Plaintext Ciphertext Key Protocol: (K, E, D) K key generation algorithm E encryp
East Los Angeles College - STUDIO - 0506
CONDITION 1PCONDITION 2PCONDITION 3PCONDITION 4PCONDITION 5PCONDITION 6PCONDITION 7PCONDITION 8PCONDITION 9PCONDITION 10PC AT H O L I CP R O T E S TA N Tindustrial shed house semi derolict building semi small business monastery
Sveriges lantbruksuniversitet - CS - 404
Public CCA-Security IntroductionCryptography and Protocols Andrei BulatovCryptography and Protocols Public CCA-Security17-2Login Problem RevisitedThe login problem Suppose that a server and a client share a secret PIN, I, that was chosen at
East Los Angeles College - STUDIO - 0506
Water Harvest Community David CookStudio 2 Strategies of Transition Belfast 2006/IndexThoughts on water, information from interview and meetings, rst semester and eld tripARw at er co m mWATER: fluid thoughtsVOLVIC EVIAN TY-NANT SAN PELL
Sveriges lantbruksuniversitet - CS - 404
Cryptography and Protocols Statistical Security5-2Symmetric Encryption SchemeA symmetric encryption scheme is a triple of algorithms (K,E,D) - K keys generation - E encryption algorithm - D decryption algorithm For simplicity assume that k K u
East Los Angeles College - STUDIO - 0506
Positive AttitudesdisplayIt is sobering to relfect, seeing the wooden-pallet towers vying with the apartment blocks and American kit-hotels - like boys standing precariously on one anothers shoulders to get ther mugs in shot - thatGroundwork NI R
Sveriges lantbruksuniversitet - CS - 404
Cryptography and Protocols Secure Channel11-2Message Authentication Schemes (Reminder)A Message Authentication Scheme (MAC) consists of 2 algorithms (Sign, Ver) There is a key shared between the signer and the verifier (Alice and Bob). Alice se
East Los Angeles College - STUDIO - 0506
ACTORSMarket Trader Residents (Catholic+Protestant) Tourists (Local+Foreign) Car owner Tour Bus Operator O.A.P Corner Shop-ownerSCENARIOPHASE 1: The local council allow some void sites around the peace walls to become sites for car-boot sales on
Sveriges lantbruksuniversitet - CS - 404
Attacks Introduction on RSACryptography and Protocols Andrei BulatovCryptography and Protocols RSA Attacks18-2Practical SchemesPKCS #1 : Public Key Cryptography Standard RSA Cryptography Standard Encryption: P plaintext, seed a random see
East Los Angeles College - STUDIO - 0506
Conflict 1 Scenario Game:The first game is played by four local actors: Local Planner Protestant resident/Local councillor Architect/Designer Local School Teacher Conflict 1 miniscenario: An employee of the Royal Victoria Hospital cannot cycle safel
Sveriges lantbruksuniversitet - CS - 404
Diffie Introduction HellmanCryptography and Protocols Andrei BulatovCryptography and Protocols Diffie Hellman15-2Key ExchangeUsing public key cryptography is expensive. A better way is to use in limited amount to generate a key for a priv
East Los Angeles College - STUDIO - 0506
material volume analysisin order to allocate appropriate storage space, this sheet will look at each material colelcted by the scheme and analyse indvidually their volumes it is also necessary to account for how these materials are transferred to a
Sveriges lantbruksuniversitet - CS - 404
Cryptography and Protocols - Probability3-2Sample Space and OutcomesExperiment s and outcomes Sample space is the set of all possible outcomes Examples - flipping a coin = {heads, tails} - flipping a pair of coins = {HH, HT, TH, TT} - horse ra
East Los Angeles College - STUDIO - 0506
5 153 42675 5471. 2. 3. 4. 5. 6.space above westlink void volume above buried weir a blocked of road creates a landlocked void space between the catholic back gardens and the peace wall space above industrial sheds a fenced off vo