We often find the probability of an event by counting the number of elements in a simple sample space.
|
|
- Virginia McDaniel
- 5 years ago
- Views:
Transcription
1 outig Methods We ofte fid the probability of a evet by coutig the umber of elemets i a simple sample space. Basic methods of coutig are: Permutatios ombiatios Permutatio A arragemet of objects i a defiite order is called a permutatio of the objects. Suppose the objects are all distict. How may differet permutatios are there?! permutatios. Uless stated otherwise, selectios are made at radom, ad all! permutatios are equally likely to occur. Each permutatio occurs with probability.! How may differet ways are there to fill the 9 battig positios from 9 players? Fid the probability that Smith will be just after Joes. 9 Homework We fill the first four battig positios from 9 players. Fid the probability that both Joes ad Smith are amog the four ad Smith is just after Joes.
2 2 ombiatios The umber of differet ways of selectig r objects out of distict objects is called the umber of combiatios of thigs take r at a time, deoted by r or. r For permutatios, the order matters: ( abc,, ) ( acb,, ). { abc,, } { acb,, } For combiatios, oly the set matters:. r! ( r)! r! r r 0 I radom selectio of distict objects, all combiatios are equally likely. A NBA team has 2 players. How may ways ca the coach choose the startig five? What is the probability that Smith will be oe of the startig five? Assume the coach picks players at radom. 5 PiclSmith [ ]. 2 A Ituitive Pick 5 amog 2. 5 PiclSmith [ ]. 2 7 PexclSmith [ ]. 2
3 3 Permutatios of No-Distict Objects { abb} How may differet ways are there of orderig three letters,,? Each letter is used oce ad oly oce. Write the two b's as b ad b2. There are 3! permutatios: a b a b b a b b b a b b b b b a b a However every sequece is idetical to aother sequece if b b 2. There are 3! 3 differet way of orderig them. 2! a b b b a b b b a The problem is idetical to that of where to place a. 3 3 ways. The sample space is simple. I geeral, How may differet ways are there of orderig 2, K objects of item, objects of item 2, objects of item K? ( ) 2!!! 2 The sample space is simple. K K!
4 4. ( aaaaabbbcc) { aaaaabbbcc} How may differet ways are there of orderig te letters,,,,,,,,,? Each letter is used oce ad oly oce. Fid the prob of,,,,,,,,,. There are 0! ways. 5!3!2! The sample space is simple, ad 5!3!2! Paaaaabbbcc [,,,,,,,,, ]. 0! A Ituitive Aswer : Pick letters oe by oe. 5 P ( a) 0 54 P ( a, a) P ( aaaaa,,,, ) P ( aaaaabbb,,,,,,, ) !3!2! P ( aaaaabbbcc,,,,,,,,, ) ! For ay permutatio, e.g., !3!2! P ( abcabcaaab,,,,,,,,, ) !
5 5. Partial Permutatio of No-Distict Objects How may differet ways are there of selectig ad orderig two letters out of { aaaaabbbcc} ( a a) the te letters,,,,,,,,,? Each letter is used at most oce. Fid P,. There are ie ways: aa ab ac ba bb bc ca cb cc However the sample space is ot simple: 5 4 aa ( ) 0 9 ab ac ba bb bc ca cb 2 cc ( ) 0 9 I geeral, No geeral rule. The probability of a outcome depeds o the experimet.
6 6. Partial Permutatio of No-Distict Objects There are 0 red balls ad 0 blue balls. We are goig to pick 0 balls ad the order them. How may differet ways are there of orderig the 0 balls? Is the sample space simple? osider 0 cells, each of which is to be filled with a ball: Sice we have 0 red balls ad 0 blue balls, each cell ca be filled with a red or blue ball. 0 There are 2 differet ways. [ ] 0 P 0 red balls 2 if the sample space were simple. However the sample space is ot simple because: P [ 0 red balls] !0! 20! 20 0 ituitive? << 2 0 We expect the probability will be highest for the evet i which the te cells are filled with five red ad five blue balls. Homework There are 0 red balls ad 0 blue balls. We are goig to pick balls ad the order them. How may differet ways are there of orderig the balls? Is the sample space simple?
7 7 ombiatios of No-Distict Objects There are 25 pairs of socks i a box. 5 pairs are red ad 0 pairs are blue. We pick seve pairs. What is the probability that there are exactly three red pairs amog the seve? Thik as if the objects are distict. Name the 25 socks as R, R,, R, B, B,, B so that the 25 socks are all distict. The there are 25 7 For pickig 3 red pairs, Therefore P[ 3 red socks] ways of selectig 7 socks, ad the sample space is simple. there are ways of selectig 3 red socks, ad there are ways of selectig 4 blue socks A Ituitive Select 7 socks oe by oe:. We wat to pick exactly three red socks. There are 7 3 combiatios of cells to which we ca place the three red socks Each outcome has the same probabiliy For example, R R R B B B B R R B R B B B , , B R B R B B R ! 8! 5! 0! 5! 0! 8!7! !3! 25! 2! 6! 2!3! 6!4! 25! 25 7
8 8 outig Permutatios of No-Distict Objects usig ombiatios Permutatios of o-distict objects ca be solved through combiatios There are 3 red shoes ad 2 blue shoes. We order the 5 shoes. How may differet ways are there of orderig the 5 shoes? What is the probability that the first two shoes are blue? Usig permutatios of o-distict objects, we ca see there are 5! 0 3!2! differet ways of orderig the five shoes. A alterative method is as follows. The problem is equivalet to selectig two cells for placig blue shoes amog the five cells. B B. There are differet ways, ad the sample space is simple. P[ first two shoes are blue ] P[ BBRRR]. 0 A Ituitive 2 blue blue P [ first two shoes are blue] 5 shoes to choose from 4 shoes to choose from 0 Homework There are 0 red balls, 0 blue balls, ad 0 white balls. How may differet ways are there of orderig the 30 balls? Is the sample space simple?
PERMUTATION AND COMBINATION
MPC 1 PERMUTATION AND COMBINATION Syllabus : Fudametal priciples of coutig; Permutatio as a arragemet ad combiatio as selectio, Meaig of P(, r) ad C(, r). Simple applicatios. Permutatios are arragemets
More informationCounting and Probability CMSC 250
Coutig ad Probabilit CMSC 50 1 Coutig Coutig elemets i a list: how ma itegers i the list from 1 to 10? how ma itegers i the list from m to? assumig m CMSC 50 How Ma i a List? How ma positive three-digit
More informationCombinatorics. Chapter Permutations. Reading questions. Counting Problems. Counting Technique: The Product Rule
Chapter 3 Combiatorics 3.1 Permutatios Readig questios 1. Defie what a permutatio is i your ow words. 2. What is a fixed poit i a permutatio? 3. What do we assume about mutual disjoitedess whe creatig
More informationCS3203 #5. 6/9/04 Janak J Parekh
CS3203 #5 6/9/04 Jaak J Parekh Admiistrivia Exam o Moday All slides should be up We ll try ad have solutios for HWs #1 ad #2 out by Friday I kow the HW is due o the same day; ot much I ca do, uless you
More information}, how many different strings of length n 1 exist? }, how many different strings of length n 2 exist that contain at least one a 1
1. [5] Give sets A ad B, each of cardiality 1, how may fuctios map A i a oe-tooe fashio oto B? 2. [5] a. Give the set of r symbols { a 1, a 2,..., a r }, how may differet strigs of legth 1 exist? [5]b.
More informationPERMUTATIONS AND COMBINATIONS
www.sakshieducatio.com PERMUTATIONS AND COMBINATIONS OBJECTIVE PROBLEMS. There are parcels ad 5 post-offices. I how may differet ways the registratio of parcel ca be made 5 (a) 0 (b) 5 (c) 5 (d) 5. I how
More informationAMC AMS AMR ACS ACR ASR MSR MCR MCS CRS
Sectio 6.5: Combiatios Example Recall our five frieds, Ala, Cassie, Maggie, Seth ad Roger from the example at the begiig of the previous sectio. The have wo tickets for a cocert i Chicago ad everybody
More information8. Combinatorial Structures
Virtual Laboratories > 0. Foudatios > 1 2 3 4 5 6 7 8 9 8. Combiatorial Structures The purpose of this sectio is to study several combiatorial structures that are of basic importace i probability. Permutatios
More informationAS Exercise A: The multiplication principle. Probability using permutations and combinations. Multiplication principle. Example.
Probability usig permutatios ad combiatios Multiplicatio priciple If A ca be doe i ways, ad B ca be doe i m ways, the A followed by B ca be doe i m ways. 1. A die ad a coi are throw together. How may results
More informationCOLLEGE ALGEBRA LECTURES Copyrights and Author: Kevin Pinegar
OLLEGE ALGEBRA LETURES opyrights ad Author: Kevi iegar 8.3 Advaced outig Techiques: ermutatios Ad ombiatios Factorial Notatio Before we ca discuss permutatio ad combiatio formulas we must itroduce factorial
More informationAMC AMS AMR ACS ACR ASR MSR MCR MCS CRS
Sectio 6.5: Combiatios Example Recall our five frieds, Ala, Cassie, Maggie, Seth ad Roger from the example at the begiig of the previous sectio. The have wo tickets for a cocert i Chicago ad everybody
More informationCOMBINATORICS 2. Recall, in the previous lesson, we looked at Taxicabs machines, which always took the shortest path home
COMBINATORICS BEGINNER CIRCLE 1/0/013 1. ADVANCE TAXICABS Recall, i the previous lesso, we looked at Taxicabs machies, which always took the shortest path home taxipath We couted the umber of ways that
More informationPERMUTATIONS AND COMBINATIONS
Chapter 7 PERMUTATIONS AND COMBINATIONS Every body of discovery is mathematical i form because there is o other guidace we ca have DARWIN 7.1 Itroductio Suppose you have a suitcase with a umber lock. The
More informationGrade 6 Math Review Unit 3(Chapter 1) Answer Key
Grade 6 Math Review Uit (Chapter 1) Aswer Key 1. A) A pottery makig class charges a registratio fee of $25.00. For each item of pottery you make you pay a additioal $5.00. Write a expressio to represet
More informationPermutation Enumeration
RMT 2012 Power Roud Rubric February 18, 2012 Permutatio Eumeratio 1 (a List all permutatios of {1, 2, 3} (b Give a expressio for the umber of permutatios of {1, 2, 3,, } i terms of Compute the umber for
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 12
EECS 70 Discrete Mathematics ad Probability Theory Sprig 204 Aat Sahai Note 2 Probability Examples Based o Coutig We will ow look at examples of radom experimets ad their correspodig sample spaces, alog
More informationlecture notes September 2, Sequential Choice
18.310 lecture otes September 2, 2013 Sequetial Choice Lecturer: Michel Goemas 1 A game Cosider the followig game. I have 100 blak cards. I write dow 100 differet umbers o the cards; I ca choose ay umbers
More informationTHE LUCAS TRIANGLE RECOUNTED. Arthur T. Benjamin Dept. of Mathematics, Harvey Mudd College, Claremont, CA Introduction
THE LUCAS TRIANLE RECOUNTED Arthur T Bejami Dept of Mathematics, Harvey Mudd College, Claremot, CA 91711 bejami@hmcedu 1 Itroductio I 2], Neville Robbis explores may properties of the Lucas triagle, a
More informationChapter 2: Probability
hapter : roaility A {FF}, B {MM}, {MF, FM, MM} The, A B 0/, B {MM}, B {MF, FM}, A B {FF,MM}, A, B a A B A B c A B d A B A B 4 a 8 hapter : roaility 9 5 a A B A B A B B A A B A B B A B B B A A c A B A B
More informationCounting III. Today we ll briefly review some facts you dervied in recitation on Friday and then turn to some applications of counting.
6.04/18.06J Mathematics for Computer Sciece April 5, 005 Srii Devadas ad Eric Lehma Lecture Notes Coutig III Today we ll briefly review some facts you dervied i recitatio o Friday ad the tur to some applicatios
More informationCh 9 Sequences, Series, and Probability
Ch 9 Sequeces, Series, ad Probability Have you ever bee to a casio ad played blackjack? It is the oly game i the casio that you ca wi based o the Law of large umbers. I the early 1990s a group of math
More information1. How many possible ways are there to form five-letter words using only the letters A H? How many such words consist of five distinct letters?
COMBINATORICS EXERCISES Stepha Wager 1. How may possible ways are there to form five-letter words usig oly the letters A H? How may such words cosist of five distict letters? 2. How may differet umber
More informationOn the Number of Permutations on n Objects with. greatest cycle length
Ž. ADVANCES IN APPLIED MATHEMATICS 0, 9807 998 ARTICLE NO. AM970567 O the Number of Permutatios o Obects with Greatest Cycle Legth k Solomo W. Golomb ad Peter Gaal Commuicatio Scieces Istitute, Uiersity
More informationShuffling Cards. D.J.W. Telkamp. Utrecht University Mathematics Bachelor s Thesis. Supervised by Dr. K. Dajani
Shufflig Cards Utrecht Uiversity Mathematics Bachelor s Thesis D.J.W. Telkamp Supervised by Dr. K. Dajai Jue 3, 207 Cotets Itroductio 2 2 Prerequisites 2 2. Problems with the variatio distace................
More information7. Counting Measure. Definitions and Basic Properties
Virtual Laboratories > 0. Foudatios > 1 2 3 4 5 6 7 8 9 7. Coutig Measure Defiitios ad Basic Properties Suppose that S is a fiite set. If A S the the cardiality of A is the umber of elemets i A, ad is
More informationUnit 5: Estimating with Confidence
Uit 5: Estimatig with Cofidece Sectio 8.2 The Practice of Statistics, 4 th editio For AP* STARNES, YATES, MOORE Uit 5 Estimatig with Cofidece 8.1 8.2 8.3 Cofidece Itervals: The Basics Estimatig a Populatio
More information2. There are n letter and n addressed envelopes. The probability that all the letters are not kept in the right envelope, is. (c)
PAGE # CHAPTER EXERCISE I. A sigle letter is selected at radom from the word PROBABILITY. The probability that the selected letter is a vowel is / / / 0. There are letter ad addressed evelopes. The probability
More informationCombinatorics and probability
Departmet of Mathematics Ma 3/03 KC Border Itroductio to Probability ad Statistics Witer 208 Lecture 3: Combiatorics ad probability Relevat textboo passages: Pitma [2]: Sectios.5.6, pp. 7 77; Appedix,
More informationChapter (6) Discrete Probability Distributions Examples
hapter () Discrete robability Distributios Eamples Eample () Two balaced dice are rolled. Let X be the sum of the two dice. Obtai the probability distributio of X. Solutio Whe the two balaced dice are
More informationRoberto s Notes on Infinite Series Chapter 1: Series Section 2. Infinite series
Roberto s Notes o Ifiite Series Chapter : Series Sectio Ifiite series What you eed to ow already: What sequeces are. Basic termiology ad otatio for sequeces. What you ca lear here: What a ifiite series
More informationShuffling. Shahrzad Haddadan. March 7, 2013
Shufflig Shahrzad Haddada March 7, 2013 Abstract I this paper we will talk about two well-kow shufflig methods, the Top to Radom ad the Riffle Shuffle. We are iterested i the umber of shuffles that will
More informationLogarithms APPENDIX IV. 265 Appendix
APPENDIX IV Logarithms Sometimes, a umerical expressio may ivolve multiplicatio, divisio or ratioal powers of large umbers. For such calculatios, logarithms are very useful. They help us i makig difficult
More informationCounting on r-fibonacci Numbers
Claremot Colleges Scholarship @ Claremot All HMC Faculty Publicatios ad Research HMC Faculty Scholarship 5-1-2015 Coutig o r-fiboacci Numbers Arthur Bejami Harvey Mudd College Curtis Heberle Harvey Mudd
More informationTable Of Contents Blues Turnarounds
Table Of Cotets Blues Turarouds Turaroud #1 Turaroud # Turaroud # Turaroud # Turaroud # Turaroud # Turaroud # Turaroud # Turaroud # Blues Turarouds Blues Soloig Masterclass Week 1 Steve Stie A Blues Turaroud
More informationSummary of Random Variable Concepts April 19, 2000
Summary of Radom Variable Cocepts April 9, 2000 his is a list of importat cocepts we have covered, rather tha a review that derives or explais them. he first ad primary viewpoit: A radom process is a idexed
More informationA generalization of Eulerian numbers via rook placements
A geeralizatio of Euleria umbers via rook placemets Esther Baaia Steve Butler Christopher Cox Jeffrey Davis Jacob Ladgraf Scarlitte Poce Abstract We cosider a geeralizatio of Euleria umbers which cout
More informationYou Think You ve Got Problems? Marc Brodie Associate Professor of Mathematics, WJU
You Thik You ve Got Problems? Marc Brodie Associate Professor of Mathematics, WJU Itroductio. My life, like that of ay other s, has its share of problems. I cosider myself fortuate, however, to have more
More informationHIGHER SECONDARY FIRST YEAR MATHEMATICS. ALGEBRA Creative Questions Time : 1.15 Hrs Marks : 45 Part - I Choose the correct answer 10 1 = 10.
www.tbtpsc.com HIGHER SEONDARY FIRST YEAR MATHEMATIS ALGEBRA eative Questios Time :. Hs Maks : Pat - I hoose the coect aswe =. The co-efficiet of middle tem i the epasio of is a) b)...( )! c).6,...( )
More information13 Legislative Bargaining
1 Legislative Bargaiig Oe of the most popular legislative models is a model due to Baro & Ferejoh (1989). The model has bee used i applicatios where the role of committees have bee studies, how the legislative
More informationNovel pseudo random number generation using variant logic framework
Edith Cowa Uiversity Research Olie Iteratioal Cyber Resiliece coferece Cofereces, Symposia ad Campus Evets 011 Novel pseudo radom umber geeratio usig variat logic framework Jeffrey Zheg Yua Uiversity,
More informationExtra Practice 1. Name Date. Lesson 1.1: Patterns in Division
Master 1.22 Extra Practice 1 Lesso 1.1: Patters i Divisio 1. Which umbers are divisible by 4? By 5? How do you kow? a) 90 b) 134 c) 395 d) 1724 e) 30 f) 560 g) 3015 h) 74 i) 748 2. Write a 5-digit umber
More informationCS 201: Adversary arguments. This handout presents two lower bounds for selection problems using adversary arguments ëknu73,
CS 01 Schlag Jauary 6, 1999 Witer `99 CS 01: Adversary argumets This hadout presets two lower bouds for selectio problems usig adversary argumets ëku73, HS78, FG76ë. I these proofs a imagiary adversary
More informationCHAPTER 5 A NEAR-LOSSLESS RUN-LENGTH CODER
95 CHAPTER 5 A NEAR-LOSSLESS RUN-LENGTH CODER 5.1 GENERAL Ru-legth codig is a lossless image compressio techique, which produces modest compressio ratios. Oe way of icreasig the compressio ratio of a ru-legth
More informationGeneral Model :Algorithms in the Real World. Applications. Block Codes
Geeral Model 5-853:Algorithms i the Real World Error Correctig Codes I Overview Hammig Codes Liear Codes 5-853 Page message (m) coder codeword (c) oisy chael decoder codeword (c ) message or error Errors
More informationVIII. Shell-Voicings
VIII. Shell-Voicigs A. The Cocept The 5th (ad ofte the root as well) ca be omitted from most 7th-chords. Ratioale: Most chords have perfect 5ths. The P5th is also preset as the rd partial i the overtoe
More informationMEI Core 2. Logarithms and exponentials. Section 2: Modelling curves using logarithms. Modelling curves of the form y kx
MEI Core 2 Logarithms ad eoetials Sectio 2: Modellig curves usig logarithms Notes ad Eamles These otes cotai subsectios o: Modellig curves of the form y = k Modellig curves of the form y = ka Modellig
More informationPlaying Leadsheets - Vol. 2
Playig Leadsheets - Vol. 2 our first volume of "Playig Leadsheets", e explored ho to play a traditioal chorale-style hym melody ritte i a leadsheet format. this issue, ill sho you ho to improvise the accompaimet
More informationDepartment of Electrical and Computer Engineering, Cornell University. ECE 3150: Microelectronics. Spring Due on April 26, 2018 at 7:00 PM
Departmet of Electrical ad omputer Egieerig, orell Uiersity EE 350: Microelectroics Sprig 08 Homework 0 Due o April 6, 08 at 7:00 PM Suggested Readigs: a) Lecture otes Importat Notes: ) MAKE SURE THAT
More information5 Quick Steps to Social Media Marketing
5 Quick Steps to Social Media Marketig Here's a simple guide to creatig goals, choosig what to post, ad trackig progress with cofidece. May of us dive ito social media marketig with high hopes to watch
More informationCombinatorics. ChaPTer a The addition and multiplication principles introduction. The addition principle
ChaPTer Combiatorics ChaPTer CoTeTS a The additio ad multiplicatio priciples b Permutatios C Factorials D Permutatios usig P r e Permutatios ivolvig restrictios F Arragemets i a circle G Combiatios usig
More informationX-Bar and S-Squared Charts
STATGRAPHICS Rev. 7/4/009 X-Bar ad S-Squared Charts Summary The X-Bar ad S-Squared Charts procedure creates cotrol charts for a sigle umeric variable where the data have bee collected i subgroups. It creates
More informationAlignment in linear space
Sequece Aligmet: Liear Space Aligmet i liear space Chapter 7 of Joes ad Pevzer Q. Ca we avoid usig quadratic space? Easy. Optimal value i O(m + ) space ad O(m) time. Compute OPT(i, ) from OPT(i-1, ). No
More informationarxiv: v2 [math.co] 15 Oct 2018
THE 21 CARD TRICK AND IT GENERALIZATION DIBYAJYOTI DEB arxiv:1809.04072v2 [math.co] 15 Oct 2018 Abstract. The 21 card trick is well kow. It was recetly show i a episode of the popular YouTube chael Numberphile.
More informationChapter 3 Digital Logic Structures
Copyright The McGraw-HillCompaies, Ic. Permissio required for reproductio or display. Computig Layers Chapter 3 Digital Logic Structures Problems Algorithms Laguage Istructio Set Architecture Microarchitecture
More informationFingerprint Classification Based on Directional Image Constructed Using Wavelet Transform Domains
7 Figerprit Classificatio Based o Directioal Image Costructed Usig Wavelet Trasform Domais Musa Mohd Mokji, Syed Abd. Rahma Syed Abu Bakar, Zuwairie Ibrahim 3 Departmet of Microelectroic ad Computer Egieerig
More informationarxiv:math/ v2 [math.pr] 14 Mar 2006
The Aals of Applied Probability 2006, Vol. 16, No. 1, 231 243 DOI: 10.1214/105051605000000692 c Istitute of Mathematical Statistics, 2006 arxiv:math/0501401v2 [math.pr] 14 Mar 2006 THE OVERHAND SHUFFLE
More informationLecture 4: Frequency Reuse Concepts
EE 499: Wireless & Mobile Commuicatios (8) Lecture 4: Frequecy euse Cocepts Distace betwee Co-Chael Cell Ceters Kowig the relatio betwee,, ad, we ca easily fid distace betwee the ceter poits of two co
More informationBlues Soloing Masterclass - Week 1
Table Of Cotets Why Lear the Blues? Tuig The Chromatic Scale ad Notes o the th Strig Notes o the th Strig Commo Notes Betwee the th ad th Strigs The I-IV-V Chord Progressio Chords i Blues Groove: Straight
More informationName Class. Date Section. Test Form A Chapter Chapter 9 Infinite Series. 1 n 1 2 n 3n 1, n 1, 2, 3, Find the fourth term of the sequence
8 Chapter 9 Ifiite Series Test Form A Chapter 9 Name Class Date Sectio. Fid the fourth term of the sequece,,,,.... 6 (a) (b) 6 (c) 8 6. Determie if the followig sequece coverges or diverges: If the sequece
More informationMathematical Explorations of Card Tricks
Joh Carroll Uiversity Carroll Collected Seior Hoors Projects Theses, Essays, ad Seior Hoors Projects Sprig 2015 Mathematical Exploratios of Card Tricks Timothy R. Weeks Joh Carroll Uiversity, tweeks15@jcu.edu
More informationDensity Slicing Reference Manual
Desity Slicig Referece Maual Improvisio, Viscout Cetre II, Uiversity of Warwick Sciece Park, Millbur Hill Road, Covetry. CV4 7HS Tel: 0044 (0) 24 7669 2229 Fax: 0044 (0) 24 7669 0091 e-mail: admi@improvisio.com
More informationOptimal Arrangement of Buoys Observable by Means of Radar
Optimal Arragemet of Buoys Observable by Meas of Radar TOMASZ PRACZYK Istitute of Naval Weapo ad Computer Sciece Polish Naval Academy Śmidowicza 69, 8-03 Gdyia POLAND t.praczy@amw.gdyia.pl Abstract: -
More information信號與系統 Signals and Systems
Sprig 2 信號與系統 Sigals ad Systems Chapter SS- Sigals ad Systems Feg-Li Lia NTU-EE Feb Ju Figures ad images used i these lecture otes are adopted from Sigals & Systems by Ala V. Oppeheim ad Ala S. Willsky,
More informationSMML MEET 3 ROUND 1
ROUND 1 1. How many different 3-digit numbers can be formed using the digits 0, 2, 3, 5 and 7 without repetition? 2. There are 120 students in the senior class at Jefferson High. 25 of these seniors participate
More informationSpread Spectrum Signal for Digital Communications
Wireless Iformatio Trasmissio System Lab. Spread Spectrum Sigal for Digital Commuicatios Istitute of Commuicatios Egieerig Natioal Su Yat-se Uiversity Spread Spectrum Commuicatios Defiitio: The trasmitted
More information15 min/ Fall in New England
5 mi/ 0+ -4 Fall i New Eglad Before witer makes its appearace, a particularly warm fall bathes the forest i a golde shimmer. Durig the Idia Summer, New Eglad blossoms oe last time. Treetops are ablaze
More informationZonerich AB-T88. MINI Thermal Printer COMMAND SPECIFICATION. Zonerich Computer Equipments Co.,Ltd MANUAL REVISION EN 1.
Zoerich AB-T88 MINI Thermal Priter COMMAND SPECIFICATION MANUAL REVISION EN. Zoerich Computer Equipmets Co.,Ltd http://www.zoerich.com Commad List Prit ad lie feed Prit ad carriage retur Trasmissio real-time
More informationMath 7 Flipped Mastery Self Tester Worksheet Name: Class:. Chapter 1 (Unit 1) Patterns and Relationships - Accommodated 1.1 Patterns In Division /36
Chapter 1 (Uit 1) Patters ad Relatioships - Accommodated 1.1 Patters I Divisio /36 Divisibility Rule Cheats; A whole umber is divisible by 2 if it is a eve umber A whole umber is divisible by 4 if the
More informationData Mining the Online Encyclopedia of Integer Sequences for New Identities Hieu Nguyen
Slide 1 of 18 Data Miig the Olie Ecyclopedia of Iteger Sequeces for New Idetities Hieu Nguye Rowa Uiversity MAA-NJ Sectio Sprig Meetig March 31, 2012 2 MAA-NJ Sprig Meetig Data Miig OEIS.b ü Ackowledgemets
More information信號與系統 Signals and Systems
Sprig 24 信號與系統 Sigals ad Systems Chapter SS- Sigals ad Systems Feg-Li Lia NTU-EE Feb4 Ju4 Figures ad images used i these lecture otes are adopted from Sigals & Systems by Ala V. Oppeheim ad Ala S. Willsky,
More informationFinal exam PS 30 December 2009
Fial exam PS 30 December 2009 Name: UID: TA ad sectio umber: This is a closed book exam. The oly thig you ca take ito this exam is yourself ad writig istrumets. Everythig you write should be your ow work.
More informationBOTTLENECK BRANCH MARKING FOR NOISE CONSOLIDATION
BOTTLENECK BRANCH MARKING FOR NOISE CONSOLIDATION IN MULTICAST NETWORKS Jordi Ros, Wei K. Tsai ad Mahadeve Iyer Departmet of Electrical ad Computer Egieerig Uiversity of Califoria, Irvie, CA 92697 {jros,
More informationMaking sure metrics are meaningful
Makig sure metrics are meaigful Some thigs are quatifiable, but ot very useful CPU performace: MHz is ot the same as performace Cameras: Mega-Pixels is ot the same as quality Cosistet ad quatifiable metrics
More informationIntroduction to Wireless Communication Systems ECE 476/ECE 501C/CS 513 Winter 2003
troductio to Wireless Commuicatio ystems ECE 476/ECE 501C/C 513 Witer 2003 eview for Exam #1 March 4, 2003 Exam Details Must follow seatig chart - Posted 30 miutes before exam. Cheatig will be treated
More informationGENERALIZED FORM OF A 4X4 STRONGLY MAGIC SQUARE
IJMMS, Vol. 1, No. Geeralized 1, (Jauary-Jue Form 016):87-9 of A 4x4 Strogly Magic Square Serials Publicatios 87 ISSN: 0973-339 GENERALIZED FORM OF A 4X4 STRONGLY MAGIC SQUARE Neeradha. C. K, ad Dr. V.
More informationApplication of Improved Genetic Algorithm to Two-side Assembly Line Balancing
206 3 rd Iteratioal Coferece o Mechaical, Idustrial, ad Maufacturig Egieerig (MIME 206) ISBN: 978--60595-33-7 Applicatio of Improved Geetic Algorithm to Two-side Assembly Lie Balacig Ximi Zhag, Qia Wag,
More information202 Chapter 9 n Go Bot. Hint
Chapter 9 Go Bot Now it s time to put everythig you have leared so far i this book to good use. I this chapter you will lear how to create your first robotic project, the Go Bot, a four-wheeled robot.
More informationDiscrete Mathematics: Logic. Discrete Mathematics: Lecture 15: Counting
Discrete Mathematics: Logic Discrete Mathematics: Lecture 15: Counting counting combinatorics: the study of the number of ways to put things together into various combinations basic counting principles
More informationPOWERS OF 3RD ORDER MAGIC SQUARES
Fuzzy Sets, Rough Sets ad Multivalued Operatios ad Applicatios, Vol. 4, No. 1, (Jauary-Jue 01): 37 43 Iteratioal Sciece Press POWERS OF 3RD ORDER MAGIC SQUARES Sreerajii K.S. 1 ad V. Madhukar Mallayya
More informationCrafting Well-Built Sentences. Varying Sentence Patterns. Breaking the Rules to Create Fluency. Capturing a Smooth and Rhythmic Flow
SENTENCE FLUENCY k e y q u a l i t i e s Craftig Well-Built Seteces Varyig Setece Patters Breakig the Rules to Create Fluecy Capturig a Smooth ad Rhythmic Flow crafting WELL-BUILT SENTENCES Do my seteces
More informationSpeak up Ask questions Find the facts Evaluate your choices Read the label and follow directions
Whe it comes to usig medicie, it is importat to kow that o medicie is completely safe. The U.S. Food ad Drug Admiistratio (FDA) judges a drug to be safe eough to approve whe the beefits of the medicie
More information20. CONFIDENCE INTERVALS FOR THE MEAN, UNKNOWN VARIANCE
20. CONFIDENCE INTERVALS FOR THE MEAN, UNKNOWN VARIANCE If the populatio tadard deviatio σ i ukow, a it uually will be i practice, we will have to etimate it by the ample tadard deviatio. Sice σ i ukow,
More informationLAB 7: Refractive index, geodesic lenses and leaky wave antennas
EI400 Applied Atea Theory LAB7: Refractive idex ad leaky wave ateas LAB 7: Refractive idex, geodesic leses ad leaky wave ateas. Purpose: The mai goal of this laboratory how to characterize the effective
More informationEMCdownload. Acknowledgements. Fair use
EMC_Sulight.idd 1 28/03/2013 09:06 Ackowledgemets Writte by Aa Sarchet, with Kate Oliver Edited by Kate Oliver Frot cover: Rebecca Scambler, 2013 Published by The Eglish ad Media Cetre, 2013 for EMCdowload.co.uk
More informationHOW BAD RECEIVER COORDINATES CAN AFFECT GPS TIMING
HOW BAD RECEIVER COORDINATES CAN AFFECT GPS TIMING H. Chadsey U.S. Naval Observatory Washigto, D.C. 2392 Abstract May sources of error are possible whe GPS is used for time comparisos. Some of these mo
More informationEfficient Feedback-Based Scheduling Policies for Chunked Network Codes over Networks with Loss and Delay
Efficiet Feedback-Based Schedulig Policies for Chuked Network Codes over Networks with Loss ad Delay Aoosheh Heidarzadeh ad Amir H. Baihashemi Departmet of Systems ad Computer Egieerig, Carleto Uiversity,
More informationCHAPTER 8 JOINT PAPR REDUCTION AND ICI CANCELLATION IN OFDM SYSTEMS
CHAPTER 8 JOIT PAPR REDUCTIO AD ICI CACELLATIO I OFDM SYSTEMS Itercarrier Iterferece (ICI) is aother major issue i implemetig a OFDM system. As discussed i chapter 3, the OFDM subcarriers are arrowbad
More informationRoom Design [ HOW TO SET UP YOUR EVENT SPACE ]
Room Desig [ HOW TO SET UP YOUR EVENT SPACE ] There are so may compoets of plaig a evet ad so may decisios! I this article you will lear about some factors that will help you choose the best space for
More informationA SELECTIVE POINTER FORWARDING STRATEGY FOR LOCATION TRACKING IN PERSONAL COMMUNICATION SYSTEMS
A SELETIVE POINTE FOWADING STATEGY FO LOATION TAKING IN PESONAL OUNIATION SYSTES Seo G. hag ad hae Y. Lee Departmet of Idustrial Egieerig, KAIST 373-, Kusug-Dog, Taejo, Korea, 305-70 cylee@heuristic.kaist.ac.kr
More informationWhat is Multiple Access? Code Division Multiple Access for Wireless Communications. Time Division Multiple Access (TDMA)
Wireless Networkig ad ommuicatios Group Wireless Networkig ad ommuicatios Group What is Multiple Access? ode Divisio Multiple Access for Wireless ommuicatios Prof. effre G. Adrews Wireless Networkig ad
More informationCross-Layer Performance of a Distributed Real-Time MAC Protocol Supporting Variable Bit Rate Multiclass Services in WPANs
Cross-Layer Performace of a Distributed Real-Time MAC Protocol Supportig Variable Bit Rate Multiclass Services i WPANs David Tug Chog Wog, Jo W. Ma, ad ee Chaig Chua 3 Istitute for Ifocomm Research, Heg
More informationIntroduction to Markov Models
Itroductio to Markov Models But first: A few prelimiaries o text preprocessig Estimatig the probability of phrases of words, seteces, etc. What couts as a word? A tricky questio. How to fid Seteces?? CIS
More informationON THE FUNDAMENTAL RELATIONSHIP BETWEEN THE ACHIEVABLE CAPACITY AND DELAY IN MOBILE WIRELESS NETWORKS
Chapter ON THE FUNDAMENTAL RELATIONSHIP BETWEEN THE ACHIEVABLE CAPACITY AND DELAY IN MOBILE WIRELESS NETWORKS Xiaoju Li ad Ness B. Shroff School of Electrical ad Computer Egieerig, Purdue Uiversity West
More informationNeighbor Discovery for Cognitive Radio Ad Hoc Networks
Neighbor Discovery for Cogitive Radio Ad Hoc Networks Zaw Htike Departmet of Computer Egieerig, Kyug Hee Uiversity, 1 Seocheo,Giheug, Yogi, Gyeoggi 449-701 Korea +8-10-561-811 htike@etworkig.khu.ac.kr,
More informationCOUNTING AND PROBABILITY
CHAPTER 9 COUNTING AND PROBABILITY Copyright Cengage Learning. All rights reserved. SECTION 9.2 Possibility Trees and the Multiplication Rule Copyright Cengage Learning. All rights reserved. Possibility
More informationConfidence Intervals. Our Goal in Inference. Confidence Intervals (CI) Inference. Confidence Intervals (CI) x $p s
Cofidece Iterval Iferece We are i the fourth ad fial part of the coure - tatitical iferece, where we draw cocluio about the populatio baed o the data obtaied from a ample choe from it. Chapter 7 1 Our
More informationH(X,Y) = H(X) + H(Y X)
Today s Topics Iformatio Theory Mohamed Hamada oftware gieerig ab The Uiversity of Aizu mail: hamada@u-aizu.ac.jp UR: http://www.u-aizu.ac.jp/~hamada tropy review tropy ad Data Compressio Uiquely decodable
More information4. INTERSYMBOL INTERFERENCE
DATA COMMUNICATIONS 59 4. INTERSYMBOL INTERFERENCE 4.1 OBJECT The effects of restricted badwidth i basebad data trasmissio will be studied. Measuremets relative to itersymbol iterferece, usig the eye patter
More informationMathematical Foundations HW 5 By 11:59pm, 12 Dec, 2015
1 Probability Axioms Let A,B,C be three arbitrary events. Find the probability of exactly one of these events occuring. Sample space S: {ABC, AB, AC, BC, A, B, C, }, and S = 8. P(A or B or C) = 3 8. note:
More informationConcurrent Fault Detection in Random Combinational Logic
Cocurret Fault Detectio i Radom Combiatioal Logic Petros Drieas ad Yiorgos Makris Departmets of Computer Sciece ad Electrical Egieerig Yale Uiversity Abstract We discuss a o-itrusive methodology for cocurret
More information