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Course: EE 1244, Fall 2010
School: Conestoga
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Word Count: 605

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VARIABLES Measurements RANDOM and observations are called random variables. Each results from an outcome of an experiment, so: Denition: Random variable X is real function on sample space S. Start with discrete r.v. RX is nite or countable. 1 For any real x RX , the set {X = x} is an event and so has a probability. Denition: The probability mass function of X is the function f (x) = P (X = x) Properties. f...

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VARIABLES Measurements RANDOM and observations are called random variables. Each results from an outcome of an experiment, so: Denition: Random variable X is real function on sample space S. Start with discrete r.v. RX is nite or countable. 1 For any real x RX , the set {X = x} is an event and so has a probability. Denition: The probability mass function of X is the function f (x) = P (X = x) Properties. f (x) 0 xRX f (x) = 1 P (X A ) = xA f (x) 2 Example: Benfords law First observed by Frank Benford working at GE Research Laboratories in 1920s. In many real-world datasets the rst digit of a number follows a distribution: First Digit Probability First Digit Probability 1 0.301 6 0.067 2 0.176 7 0.058 3 0.125 8 0.051 4 0.097 9 0.046 5 0.079 Consider a market capital of a randomly picked publicly traded company. Let X = rst digit of market capital amount. According to Benfords law P (X = 1) = 0.301. References: Ted Hill, The rst-digit phenomenon, American Scientist, 86(1996), pp.358-363. Fraud detection: http://www.nigrini.com/ 3 Origin of Benfords Law Leading digit X of positive numbers is the rst non-zero number in its decimal representation. So the leading digits of .0023, 2.23, and .234 are all 2. Benford wasnt the rst to observe this phenomenon. Astronomer Simon Newcomb, looking at tables of logarithm, concluded that leading distribution of real data was not typically uniformly distributed. In 1881 he gave an argument to show that the mantissas of the logarithms of the numbers should be uniform P (X = x) = f (x) = log(x + 1) log(x) for x = 1, 2, .., 9 x f ( x) 1 .301 2 .176 3 .125 4 .097 5 .079 6 .067 7 .058 8 .051 4 9 .046 Example 2.3-1, p 107. Fair coin tossed three times. X = # heads. 3 P (X = x ) = x x 0 1 2 3 13 2 f (x) 1/8 3/8 3/8 1/8 sum = 1 5 Example 2.3-2, p 108. Toss fair coin repeatedly until rst head. X = # tosses. 11 11 1x P (X x) = = = ... 22 22 2 x 1 times Verify 1x x=1 2 = 1 using sum of ge- ometric series. Denition: The (cumulative) distribution function of X is the function F (x) P (X x) = f (t) tx 6 Use either f (x) or F (x) to nd probabilities. For Example 2.3-2, nd P (X = 3, 4, 5). P (X = 3, 4, 5) = = f (x) x=3,4,5 13 2 14 15 7 + + = 2 2 32 OR rst nd 1t 1x F (x) = =1 2 t=1 2 x and recognize that P (a X b) = F (b) F (a 1) 7 Then P (X = 3, 4, 5) = P (X 5) P (X 2) 15 12 =1 (1 ) 2 2 1 7 1 = = 4 32 32 8 Expectations Expectation (or expected value or mean value or mean) is the statistical equivalent of an average. Multiple each possible value of a r.v. by its probability mass function. E [X ] = xf (x) x More generally if u is a real function then u(X ) is also a random variable and its expectation is E [u(X )] = u(x)f (x) x 9 E [X ] is often denoted by X or simply if no ambiguity. Expectation has many properties. It is the centre of mass of the probability mass function. Also, if X1, X2, . . . , XN are observations made of a measurement X with mean then the long run average value is approximately E [X ] for large N , that is X1 + X2 + . . . + XN N This is called the Law of Large Numbers or more colloquially, the law of averages. 10 Find the expected value of Lotto. Earlier x 0 1 2 3 4 5 6 f (x) 0.436 0.413 0.132 0.0177 0.000969 0.0000184 0.0000000715 6 E [X ] = xf (x) = 0.734 x=1 So on average you lose about 27 cents for each dollar wagered. 11 Variance and Standard Deviation Denition: Variance of X is given by VarX 2 = E [(X )2] = (x)2f (x) x Denition: Standard deviation of X is , the positive square root of the variance. Equivalent computational formula 2 = E [X 2] 2 12 We will see shortly that is a measure of the variability of a random variable and plays for X a similar role that s plays for a data set. Exercise 2.3-5. To be done in class. 13
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Conestoga - EE - 1244
Binomial DistributionAssumptions n independent trials trial i has dichotomous outcome (0 or 1, no or yes, failure or success, etc.) denotedby Xi. Called Bernoulli trials. probability of success on anytrial is the same p.1Let Y = # of successes.Y
Conestoga - EE - 1244
Example:- 19 19 square Go board (Japanese boardgame)- throw dart repeatedly and at random- mark square hit111. P (specic square hit) = 19 1= 1193612. How many throws, until highprobability of hitting some squareagain?3. After N throws, what
Conestoga - EE - 1244
Poisson DistributionFlaws occur at random alonglength of oil pipeline. Average per unit length. Y = number ofaws in a randomly selected section of pipeline of length 1. Canwe determine the distribution ofY?Generic situation is incidentshappening o
Conestoga - EE - 1244
MULTIVARIATEDISTRIBUTIONS, DATARarely only one random variable.Usually many cfw_X1, . . . , Xn &amp; function u(X1, . . . , Xn).distribution.Need jointn = 2. Notationcfw_X, Y . Joint p.m.f.f (x, y ) = P (X = x and Y = y ) P (X = x, Y = y )1.Exerci
Conestoga - EE - 1244
Continuous RandomVariablesContinuous random variables are random variables that arise from measurements can take any value in intervalExample.Random numbers in [0, 1]used in random number generators.1.Think of a circle with circumference 1 and a
Conestoga - EE - 1244
NORMAL DistributionMost important distribution in statistics112f (x) = e 2x 21E [X ] =Var X =P (X x) = F (x)21x 1 t= t= e 2 dt 2No formula for this integral. Tables exist C4 p568-9 for standard normalStandard Normal = 0, = 1.Notation:
Conestoga - EE - 1244
Multiple Continuous RandomVariablesJoint probability density functionfX,Y (x, y ) describes pair (X, Y ).fX,Y (x, y ) 0f(x, y )dydxx= y = X,Y=1P (X, Y ) A) =Cumulative joint distribution functionFX,Y (x, y ) =1Marginal DensitiesThe p.d.f. of
Conestoga - EE - 1244
Data and Random VariablesAt the start of this course we discussed variation in data. Suppose that there are two methods of making concrete aggregates, an existingone (denoted by symbol A), and a new one,(denoted by B ), which contains a new bindingage
Conestoga - EE - 1244
Central Limit TheoremNotation. N (, 2) refers to normal with mean and variance 2.Have just seen that when X1, . . . , Xn is a random sample from N (, 2) then sample meansX N (, 2/n). This follows from a more general result that:2If Xi N (i, i ) are
Conestoga - EE - 1244
Introduction to EstimationData arise from measurements whose valuesdepend on characteristics of the process or experiment being considered. Some features ofthe process can be controlled while others cannot and result in variation in the data. If wecou
Conestoga - EE - 1244
Condence IntervalsDenition. Level C condence interval (C.I.) for is interval (L, U ) whereL, U obtained from data, such thatP (L U ) = Cno matter what .Typically, C = 0.90, 0.95, 0.99.C.I. is interval estimator, represents setof plausible estimates
Conestoga - EE - 1244
Condence Intervals for 1 2 Compare responses in two groups Can arise in two ways1.2.12Population12SampleSizen1n2VariableMeanSt. dev.XY1212Parameter of interest=Natural estimator=ThenE[X Y ] =3Assume independent samples. Then
Conestoga - EE - 1244
C.I. for from NormalPopulation with Unknown Previously assumed either:Now is unknown, but sample sizenot large enough for central limit theorem. Inference needs some additionalassumptions.Assume N (, 2) population. Intereston parameter but unknown.
Conestoga - EE - 1244
This following material covers Section 4.4: condence intervals forproportions and variances. Youare not responsible for the derivation (shown in purple) of thesecondence intervals on a test ornal exam. I have shown thedetails for those who are intere
Conestoga - EE - 1244
Testing a Single ParameterSignicance tests are dierent type of statistical inference. Want to choose between competing hypotheses about a parameter. Denition:An hypothesis is an assertionabout a population or about a parameter.We distinguish two kind
Conestoga - EE - 1244
Using OC Curve to DetermineSample SizeExample 4.5-1, p 255. Standard production setup. Average time = 30minutes to complete task. Assume X N (30, 1). Change suggested. New X N (, 1) where hoped &lt; 30. Samplesize n workers to testH0 : = 30H1 : &lt; 30
Conestoga - EE - 1244
Testing a Normal MeanEarlier found sample size to achievespecied Type I error and TypeII error at particular value ofalternative parameter.Here ngiven. Cant pick it. Stuck withwhatever emerges. Test at thelevel of signicance H0 : = 0H 1 : &lt; 01
Conestoga - EE - 1244
Tests Comparing 1 with 2Independent random samples cfw_X1, . . . , Xn1 and22cfw_Y1, . . . , Yn2 from N (1, 1 ), N (2, 2 ) populations respectively. Null hypothesis is:H0 : 1 = 222If 1 , 2 known then test based on normal standardized scoreZ=22I
Conestoga - EE - 1244
Paired t-testEarlier tested 1 = 2 using two independent random samples cfw_X1, . . . , Xn1 and cfw_Y1, . . . , Yn2 from22N (1, 1 ), N (2, 2 ) populations.Basis:X Y N 1 221,n122 +n22-sample Z or 2-sample t, dependingon whether variances w
Conestoga - EE - 1244
Paired t-testEarlier tested 1 = 2 using two independent random samples cfw_X1, . . . , Xn1 and cfw_Y1, . . . , Yn2 from22N (1, 1 ), N (2, 2 ) populations.Basis:X Y 2-sample Z or 2-sample t, dependingwhether variances were known or not.1Varianc
Conestoga - EE - 1244
Testing Equality of Two Proportionsp1 = p2Testing H0 : p1 = p2. = p1 p 2Independent samples Y1, Y2.Y1 binomial(n1, p1), Y2 binomial(n2, p2)YY = P1 P2 = n1 n212Large sample sizes n1, n2.Use normal approximation.Pi N pi, pi(1pi , i = 1, 2ni N
Conestoga - EE - 1244
Conestoga - EE - 1244
Engineering Design &amp; GraphicsEngineering 1C03Dr. Thomas E. Doyle, P.EngDepartment of Electrical and Computer EngineeringETB/106Ofce Hours are Mondays 2:00 - 4:00pmWelcome!IntroductionOverviewPeopleSyllabusSoftwareLecture, Laboratory, Tutorial
Conestoga - EE - 1244
Introduction toTechnical Sketches andEngineering DrawingsDr. Thomas E. Doyle, P.EngDepartment of Electrical and Computer EngineeringETB/106Ofce Hours are Mondays 2:00 - 4:00pmLecture Objectives Geometry of Engineering Graphics Dene projection typ
Conestoga - EE - 1244
Engineering Design &amp;Graphics Week 04Dr. T. E. DoyleOverview Continue example from last lecture Thanksgiving holiday and makeup labs/tutorials Rigid bodies in free space (3-Drelationships)Thanksgiving ReschedulingIf you normally have a lab or tut
Conestoga - EE - 1244
Engineering Design &amp;Graphics Week 06Dr. T. E. DoyleAnnouncements Exams - this week! (worth 10% each) Project groups - must be decided for nextweek Maple/MapleSim - installed for next week Software available on campus in KTH/B123OverviewContinue
Conestoga - EE - 1244
Simple Mechanisms andSystem ModellingEngineering Design &amp; GraphicsEngineering 1C03Dr. T. E. DoyleOverview Simple Mechanisms Introduce System Modelling via MapleSim Introduce ProjectMechanicsMechanics deals with motion, time, and forces.M
Conestoga - EE - 1244
Introduction to ProjectProduct &amp; MapleSimEngineering Design &amp; GraphicsEngineering 1C03Dr. T. E. DoyleOverview Review project specication (noteextended deadline for part-1) Discuss the product, disassemble, reviewmechanism Introduce System
Conestoga - EE - 1244
Introduction to GearDesign: Spur &amp; Rack GearsEngineering Design &amp; Graphics Engineering 1C03 Dr. T. E. Doyle Overview Course Evaluation Project Update Introduce Ideal Spur Gear Design Design Equations Inventor Design Accelerator
Conestoga - EE - 1244
McMaster University Engineering Design &amp; Graphics Engineering 1C03 Dr. T. E. Doyle, P.Eng Dept. of Electrical and Computer Engineering Overview Recall spur gear terminology Examples of gear ratios Gear design formulae
Conestoga - EE - 1244
Ideal Worm Gear Design and Modelling McMaster University Engineering Design &amp; Graphics Engineering 1C03 Dr. T. E. Doyle, P.Eng Dept. of Electrical and Computer Engineering Worm Arrangements Worm Worm Ge
Conestoga - EE - 1244
Course ConclusionDesign &amp; GraphicsEngineering 1Dr. T. E. DoyleWeek Overview Final Exam Announcement DuraBon: 2 hours Material includes topics covered from week 1 through week 12, inclusive There will be no quesBons rega
Conestoga - EE - 1244
Introduction to Professional Engineering Lecture 1COURSE &amp; PROJECT INTRODUCTIONENG 1P03 Lecture 1September 13/15, 2010GREATEST ENGINEERING ACHIEVEMENTS OF THE 20TH CENTURYElectrificationAutomobileAirplaneWater Supply andDistributionElectronicsR
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Engineers WithoutBorders McMasterFirst General MeetingThursday, September 23rd7 8 PM in JHE 328 (Grad Lounge)ORFriday, September 24th1:30 2:30 PM in JHE A114Introduction to Professional Engineering Lecture 2DESIGN PROCESS &amp; BRAINSTORMINGENG 1P03
Conestoga - EE - 1244
McMaster Engineering SocietyWho we are, and what we do!Monday September 27thWednesday September 29th1P03 Lecture PresentationMcMaster Engineering SocietyOur MissionThe McMaster Engineering Society will fosterthe development of well roundedundergr
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Research andProject ManagementIntroduction to Professional Engineering Lecture 4OUTLINE1. Level 1 Information Session (SallyWilliams)2. Finding Information (Linda Michtics)3. Project Management4. Patents5. Apollo 13October 4/6, 2010ENG 1P03 Lec
Conestoga - EE - 1244
FUNCTIONSIntroduction to ProfessionalEngineering Lecture 5LECTURE OUTLINE1.2.3.4.AnnouncementsFunctionsDesign ExampleControl TheoryOct. 13 &amp; 18ENG 1P03 2010 Lecture 52WHAT TO TAKE NOTES ON Functions: Definitions of function, means, level
Conestoga - EE - 1244
UsersIntroduction to Professional Engineering Lecture 6Oct. 20 &amp; 25, 2010ENG 1P03 Lecture 61Outline1.2.3.4.AnnouncementsThe CamaroHuman Participant Research EthicsDesign Example (revisited)Oct. 20 &amp; 25, 2010ENG 1P03 Lecture 62ANNOUNCEMENT
Conestoga - EE - 1244
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ENGINEERINGSCIENCEAnnouncements1. Midterm exam marks will be up November 4th.2. ETB ToursNov. 3 &amp; 8, 2010ENG 1P03 2010 Lecture 82Keep/discard/improveNov. 3 &amp; 8, 2010ENG 1P03 2010 Lecture 83WHERE DOES THE MATH ANDSCIENCE COME IN?ENG 1P03 2009
Conestoga - EE - 1244
Outline1.2.3.4.5.6.7.AnnouncementsCourse EvaluationAssignment 2SustainabilityLCAWhat are engineers doing?Eco-effectivenessNov. 10 &amp; 15, 2010ENG 1P03 2010 Lecture 92Nov. 10 &amp; 15, 2010ENG 1P03 2010 Lecture 93ETB Tour Lots of spaces sti
Conestoga - EE - 1244
Outline1. Announcements2. LCA3. What are engineers doing today withrespect to sustainability?4. Eco-effectiveness5. Professionalism Exercise6. The ProfessionNov. 17 &amp; 22, 2010ENG 1P03 2010 Lecture 102ANNOUNCEMENTSNov. 17 &amp; 22, 2010ENG 1P03 20
Conestoga - EE - 1244
Outline1. Announcements2. PEO Code of Ethics3. Safety1. Ford Pinto2. Volvo SCC3. ABSNov. 24 &amp; 29ENG 1P03 2010 Lecture 112ANNOUNCEMENTSNov. 24 &amp; 29ENG 1P03 2010 Lecture 113Engineering 1 ShowcaseTuesday, December 7th from 10 AM to 12:30 PMin
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BP &amp; ReviewIntroduction to Professional EngineeringLecture 12OUTLINE1.2.3.4.AnnouncementsBPFinal Exam ReviewPath of Engineering GraduatesDec. 1 &amp; 6ENG 1P03 2010 Lecture 122ANNOUNCEMENTSDec. 1 &amp; 6ENG 1P03 2010 Lecture 123ENGINEERING 1 SH
Abraham Baldwin Agricultural College - ACC - 284
C HAPTER88-1.SUBSTANTIVE TESTSOF RECEIVABLES AND SALESTests of details of financial balances are designed to determine the reasonablenessof the balances in sales, accounts receivable, and other account balances which areaffected by the sales and co
Abraham Baldwin Agricultural College - ACC - 284
CHAPTER1OVERVIEW OF THEAUDIT PROCESS1-1. Auditors reports are important to users of financial statements because they informusers of the auditors opinion as to whether or not the statements are fairly statedor whether no conclusion can be made with
Abraham Baldwin Agricultural College - ACC - 284
CHAPTER2AUDIT PLANNING2-1. Audit risk: The risk that the auditor may unknowingly fail to appropriately modifyhis/her opinion of financial statements that are materially misstated.Inherent risk: Relates to the susceptibility of an account balance or c
Abraham Baldwin Agricultural College - ACC - 284
CHAPTER3AUDIT OF THE REVENUE AND COLLECTIONCYCLE: TESTS OF CONTROLS ANDSUBSTANTIVE TESTS OF TRANSACTIONS3-1. Directly. Higher levels of control risk induce auditors to audit larger samples ofreceivables, with confirmation date closer to the fiscal y
Abraham Baldwin Agricultural College - ACC - 284
CHAPTER4AUDIT OF THE EXPENDITURE CYCLE:TESTS OF CONTROLS AND SUBSTANTIVE TESTSOF TRANSACTIONS - I4-1. Transactions in the expenditure cycle are recorded in the purchases and cash paymentsjournal or in the voucher register and the check register. Rel
Abraham Baldwin Agricultural College - ACC - 284
CHAPTER5AUDIT OF THE EXPENDITURE CYCLE:TESTS OF CONTROLS AND SUBSTANTIVE TESTSOF TRANSACTIONS - II5-1. The following duties must be separated in the preparation of payroll: hiring, reportingand approval of time, paycheck preparation, check signing,
Abraham Baldwin Agricultural College - ACC - 284
CHAPTER6AUDIT OF THE FINANCING AND INVESTINGCYCLE: TESTS OF CONTROLS ANDSUBSTANTIVE TESTS OF TRANSACTIONS6-1. Investing activities include an entitys activities to invest in debt or equity securities ofother entities and investments in property, pla
Abraham Baldwin Agricultural College - ACC - 284
CHAPTER7SUBSTANTIVE TESTSOF CASH7-1. The quoted statement is not accurate. In their work on cash, auditors are primarilyconcerned with the risk of an overstatement of the cash balance. The listing of anon-existent or fictitious check on the outstand
Abraham Baldwin Agricultural College - ACC - 284
CHAPTER8SUBSTANTIVE TESTSOF RECEIVABLES AND SALES8-1. Tests of details of financial balances are designed to determine the reasonableness ofthe balances in sales, accounts receivable, and other account balances which areaffected by the sales and col
Abraham Baldwin Agricultural College - ACC - 284
CHAPTER9SUBSTANTIVE TESTSOF INVENTORIES ANDCOST OF GOODS SOLD9-1. Substantiation of the figure for inventories is an especially challenging task because ofthe variety of acceptable methods of valuation. In addition, the variety ofmaterials found in
Abraham Baldwin Agricultural College - ACC - 284
CHAPTER10SUBSTANTIVETESTSOFINVESTMENTS10-1.The CPAs would accept a confirmation of the securities on hand from thecustodian in lieu of their personal inspection of the securities after they hadinvestigated and satisfied themselves as to the standin
Abraham Baldwin Agricultural College - ACC - 284
CHAPTER11SUBSTANTIVE TESTS OFPROPERTY, PLANT ANDEQUIPMENT11-1.Factors which facilitate the auditors verification of plant and equipment but arenot applicable to audit work on current assets include the following:(a) High peso amount of individual
Abraham Baldwin Agricultural College - ACC - 284
CHAPTER1212-1.SUBSTANTIVE TESTS OFINTANGIBLE ASSETSThe decision whether a given expenditure on intangible asset to be treated asexpense or asset requires judgment. Expenditure giving rise to future benefits willbe classified as assets while those e
Abraham Baldwin Agricultural College - ACC - 284
CHAPTER13SUBSTANTIVE TESTS OFPREPAID EXPENSES ANDDEFERRED CHARGES13-1.Rights and obligations are tested by examining the insurance policies andconfirming the policy with insurance carriers. In turn, an auditor tests valuationby recalculating unexp
Abraham Baldwin Agricultural College - ACC - 284
CHAPTER1414-1.SUBSTANTIVE TESTS OFLIABILITIESa.Accounts receivable the auditors objective is to test the existence ofaccounts receivable.Accounts payable the auditors objective is to test the completeness ofaccounts payable.b.Accounts receivabl
Abraham Baldwin Agricultural College - ACC - 284
CHAPTER1515-1.SUBSTANTIVE TESTS OFOWNERS EQUITY ACCOUNTS(a) Procedures applicable to the existence or occurrence of shareholders equitybalances are: (1) review authorizations and terms of share issues, (2) confirmshares outstanding with registrar a
Abraham Baldwin Agricultural College - ACC - 284
CHAPTERSUBSTANTIVE TESTS OFINCOME STATEMENT ACCOUNTS1616-1.a16-2.d16-3.Red CompanyRequirement (1)2006Reported net incomeSubtract ending inventory overstatementAdd beginning inventory overstatementSubtract wages payable when incurredAdd wag