《Chap 3 Probability - Faculty of Biomedical Engineering, Iran 第一章3概率-生物医学工程学院伊朗》由会员分享，可在线阅读，更多相关《Chap 3 Probability - Faculty of Biomedical Engineering, Iran 第一章3概率-生物医学工程学院伊朗（159页珍藏版）》请在金锄头文库上搜索。
1、3 - 3 - 1 1 2003 Pearson Prentice Hall 2003 Pearson Prentice HallProbability3 - 3 - 2 2 2003 Pearson Prentice Hall 2003 Pearson Prentice HallExperiments, Outcomes, & Events3 - 3 - 3 3 2003 Pearson Prentice Hall 2003 Pearson Prentice HallExperiments & Outcomes1. Experimentn nProcess of Obtaining an O
2、bservation, Process of Obtaining an Observation, Outcome or Simple EventOutcome or Simple Event2. Sample Space (S) n nCollection of Collection of AllAll Possible Outcomes Possible Outcomes3 - 3 - 4 4 2003 Pearson Prentice Hall 2003 Pearson Prentice HallOutcome ExamplesToss a Coin, Note FaceHead, Tai
3、lToss 2 Coins, Note Faces HH, HT, TH, TTPlay a Football GameWin, Lose, TieInspect a Part, Note QualityDefective, OKObserve GenderMale, FemaleExperimentSample Space3 - 3 - 5 5 2003 Pearson Prentice Hall 2003 Pearson Prentice HallEventsAny Collection of Sample Points (outcomes)Simple Eventn nCollectio
4、n of outcomes thats simple to Collection of outcomes thats simple to describedescribeCompound Eventn nCollection of outcomes that is described Collection of outcomes that is described as unions or intersections of other eventsas unions or intersections of other events3 - 3 - 6 6 2003 Pearson Prentic
5、e Hall 2003 Pearson Prentice HallEvent ExamplesSample SpaceHH, HT, TH, TT1 Head & 1 TailHT, THHeads on 1st CoinHH, HTAt Least 1 HeadHH, HT, THHeads on BothHHExperiment: Toss 2 Coins. Note Faces.EventOutcomes in Event3 - 3 - 7 7 2003 Pearson Prentice Hall 2003 Pearson Prentice HallSample Space3 - 3 -
6、 8 8 2003 Pearson Prentice Hall 2003 Pearson Prentice HallVisualizing Sample Space1.Listingn nS = Head, TailS = Head, Tail2.Venn Diagram 3.Contingency Table4.Decision Tree Diagram3 - 3 - 9 9 2003 Pearson Prentice Hall 2003 Pearson Prentice HallSTailHHHHTTTTTHTHHTHTSample SpaceSample SpaceS = HH, HT,
7、 TH, TTS = HH, HT, TH, TTVenn DiagramOutcomeOutcomeExperiment: Toss 2 Coins. Note Faces.Event Event 3 - 3 - 1010 2003 Pearson Prentice Hall 2003 Pearson Prentice Hall2 2ndnd Coin Coin1 1stst Coin CoinHeadHeadTailTailTotalTotalHeadHeadHHHHHTHTHH, HTHH, HTTailTailTHTHTTTTTH, TTTH, TTTotalTotalHH,HH, T
8、HTH HT,HT, TTTTS SContingency TableExperiment: Toss 2 Coins. Note Faces.S = HH, HT, TH, TTS = HH, HT, TH, TTSample SpaceSample SpaceOutcomeOutcomeSimpleSimpleEvent Event (Head on(Head on1st Coin)1st Coin)3 - 3 - 1111 2003 Pearson Prentice Hall 2003 Pearson Prentice HallTree DiagramOutcome Outcome S
9、= HH, HT, TH, TTS = HH, HT, TH, TTSample SpaceSample SpaceExperiment: Toss 2 Coins. Note Faces.THTHTHHHTTHTTH3 - 3 - 1212 2003 Pearson Prentice Hall 2003 Pearson Prentice HallProbabilities3 - 3 - 1313 2003 Pearson Prentice Hall 2003 Pearson Prentice HallWhat is Probability?1.1. Numerical Measure Num
10、erical Measure of Likelihood that of Likelihood that Event Will OccurEvent Will Occurn nP P(Event)(Event)n nP P(A)(A)n nProbProb(A(A) )2.2. Lies Between 0 & 1Lies Between 0 & 13.Sum of outcome Sum of outcome probabilities is 1probabilities is 11 1.5 .5 0 0CertainCertainImpossibleImpossible3 - 3 - 14
11、14 2003 Pearson Prentice Hall 2003 Pearson Prentice HallProbabilityP(A)=limn(A)/N0N03 - 3 - 1515 2003 Pearson Prentice Hall 2003 Pearson Prentice HallMany Repetitions!Number of TossesNumber of TossesTotal Heads /Total Heads /Number of TossesNumber of Tosses0.000.000.250.250.500.500.750.751.001.000 0
12、2525505075751001001251253 - 3 - 1616 2003 Pearson Prentice Hall 2003 Pearson Prentice HallConditional Probability3 - 3 - 1717 2003 Pearson Prentice Hall 2003 Pearson Prentice HallConditional Probability1. Event Probability Given that Another Event Occurred2. Revise Original Sample Space to Account f
13、or New Informationn nEliminates Certain OutcomesEliminates Certain Outcomes3.P(A | B) = P(A and B) P(B)3 - 3 - 1818 2003 Pearson Prentice Hall 2003 Pearson Prentice HallSBlackAceConditional Probability Using Venn DiagramBlack Happens: Black Happens: Eliminates All Eliminates All Other OutcomesOther
14、OutcomesEvent (Ace AND Black)Event (Ace AND Black)(S)Black3 - 3 - 1919 2003 Pearson Prentice Hall 2003 Pearson Prentice HallColorColorTypeTypeRedRedBlackBlackTotalTotalAceAce2 22 24 4Non-AceNon-Ace242424244848TotalTotal262626265252Conditional Probability Using Contingency TableExperiment: Draw 1 Car
15、d. Note Kind, Color & Suit.Revised Revised Sample Sample SpaceSpace3 - 3 - 2020 2003 Pearson Prentice Hall 2003 Pearson Prentice HallStatistical Independence1.1.Event Occurrence Event Occurrence Does Does NotNot Affect Probability of Another Event Affect Probability of Another Eventn nP(P(A A | B) =
16、 P( | B) = P(A A) )Example: Toss 1 Coin Twice (independent)Example: Toss 1 Coin Twice (independent)n nP(second toss H)= P(second toss H)= n nP(second toss H | first toss H) = P(second toss H | first toss H) = 3 - 3 - 2121 2003 Pearson Prentice Hall 2003 Pearson Prentice HallTree DiagramExperiment: S
17、elect 2 Pens from 20 Pens: 14 Blue & 6 Red. Dont Replace.Dependent!Dependent!BRBRBRP(R) = 6/20P(R) = 6/20P(R|R) = 5/19P(R|R) = 5/19P(B|R) = 14/19P(B|R) = 14/19P(B) = 14/20P(B) = 14/20P(R|B) = 6/19P(R|B) = 6/19P(B|B) = 13/19P(B|B) = 13/193 - 3 - 2222 2003 Pearson Prentice Hall 2003 Pearson Prentice H
18、allThinking ChallengeUsing the Table Then the Formula, Whats the Probability?Pr(C)=P(B|C) =P(C|B) = Are C & B Independent?EventEventEventEventC CD DTotalTotalA A4 42 26 6B B1 13 34 4TotalTotal5 55 510103 - 3 - 2323 2003 Pearson Prentice Hall 2003 Pearson Prentice HallSolution*Using the Formula, the
19、Probabilities Are:DependentDependentP(C | B) = P(C B)P(B)P(C) = 5 510101 104 10141 14 4/3 - 3 - 2424 2003 Pearson Prentice Hall 2003 Pearson Prentice HallMultiplicative Rule3 - 3 - 2525 2003 Pearson Prentice Hall 2003 Pearson Prentice HallMultiplicative Rule1. Used to Get Compound Probabilities for
20、Intersection of Eventsn nCalled Joint EventsCalled Joint Events2. P(A and B) = P(A B)= P(A)*P(B|A) = P(B)*P(A|B)3. For Independent Events:P(A and B) = P(A B) = P(A)*P(B)3 - 3 - 2626 2003 Pearson Prentice Hall 2003 Pearson Prentice HallMultiplicative Rule ExampleExperiment: Draw 1 Card. Note Kind, Co
21、lor & Suit.ColorColorTypeTypeRedRedBlackBlackTotalTotalAceAce2 22 24 4Non-AceNon-Ace242424244848TotalTotal2626262652523 - 3 - 2727 2003 Pearson Prentice Hall 2003 Pearson Prentice HallThinking ChallengeUsing the Multiplicative Rule, Whats the Probability?P(C B) =P(B D) =P(A B) =EventEventEventEventC
22、 CD DTotalTotalA A4 42 26 6B B1 13 34 4TotalTotal5 55 510103 - 3 - 2828 2003 Pearson Prentice Hall 2003 Pearson Prentice HallSolution*Using the Multiplicative Rule, the Probabilities Are:P(C P(C B) B) = = P(C) P(C) P(B|P(B|C) = 5/10 * 1/5 = 1/10 C) = 5/10 * 1/5 = 1/10 P(B P(B D) D) = = P(B) P(B) P(D
23、|P(D|B) = 4/10 * 3/4 = 3/10 B) = 4/10 * 3/4 = 3/10 P(A P(A B) B) = = P(A) P(A) P(B|P(B|A)A) 0 0 3 - 3 - 2929 2003 Pearson Prentice Hall 2003 Pearson Prentice HallIndependence RevisitedIf A is independent of B, B is independent of AIf A is independent of B, B is independent of An nP(A and B) = P(B|A)
24、P(A)=P(A|B)P(B)P(A and B) = P(B|A)P(A)=P(A|B)P(B)n nP(A|B)=P(A) P(A|B)=P(A) P(B|A)P(A) = P(A)P(B) P(B|A)P(A) = P(A)P(B) P(B|A)=P(B) P(B|A)=P(B)Equivalence of the two independence definitions:Equivalence of the two independence definitions:P(A and B) = P(A)*P(B) if and only if P(B|A) = P(B)P(A and B)
25、 = P(A)*P(B) if and only if P(B|A) = P(B)n nP(A and B) = P(A)P(B|A)P(A and B) = P(A)P(B|A)n nIf P(B|A) = P(B), then P(A and B) = P(A)P(B)If P(B|A) = P(B), then P(A and B) = P(A)P(B)n nIf P(B|A) != P(B), then P(A and B) != P(A)P(B)If P(B|A) != P(B), then P(A and B) != P(A)P(B)3 - 3 - 3030 2003 Pearso
26、n Prentice Hall 2003 Pearson Prentice HallRandom Variable3 - 3 - 3131 2003 Pearson Prentice Hall 2003 Pearson Prentice HallRandom VariablesA random variableA random variable ( (rvrv) ) X X is a mapping (function) from the is a mapping (function) from the sample space sample space S S to the set of r
27、eal numbers to the set of real numbersn n If If image(image(X X ) ) finite or countable infinite, finite or countable infinite, X X is a discrete is a discrete rvrvInverse image of a real number Inverse image of a real number x x is the set of all sample points is the set of all sample points that a
28、re mapped by that are mapped by X X into into x x: :It is easy to see that It is easy to see that 3 - 3 - 3232 2003 Pearson Prentice Hall 2003 Pearson Prentice HallDiscrete Random Variable: pmfpk3 - 3 - 3333 2003 Pearson Prentice Hall 2003 Pearson Prentice HallDiscrete Random Variable: CDF3 - 3 - 34
29、34 2003 Pearson Prentice Hall 2003 Pearson Prentice HallProbability Mass Function (pmf)A Ax x : set of all sample points such that,: set of all sample points such that, pmfpmf 3 - 3 - 3535 2003 Pearson Prentice Hall 2003 Pearson Prentice Hallpmf Properties Since a discrete Since a discrete rvrv X ta
30、kes a finite or a X takes a finite or a countablycountably infinite set infinite set values, values, the last property above can be restated as, the last property above can be restated as,3 - 3 - 3636 2003 Pearson Prentice Hall 2003 Pearson Prentice HallDistribution Functionpmfpmf: defined for a spe
31、cific : defined for a specific rvrv value, i.e., value, i.e.,Probability of a set Probability of a set n n Cumulative Distribution Function (CDF)Cumulative Distribution Function (CDF) 3 - 3 - 3737 2003 Pearson Prentice Hall 2003 Pearson Prentice HallDistribution Function properties 3 - 3 - 3838 2003
32、 Pearson Prentice Hall 2003 Pearson Prentice HallEquivalence: n n Probability mass function Probability mass functionn n Discrete density function Discrete density function(consider integer valued random variable)(consider integer valued random variable)cdf:pmf: Discrete Random Variables3 - 3 - 3939
33、 2003 Pearson Prentice Hall 2003 Pearson Prentice HallCommon discrete random variables Constant Constant UniformUniformBernoulliBernoulliBinomialBinomialGeometricGeometricPoissonPoissonExponentialExponential3 - 3 - 4040 2003 Pearson Prentice Hall 2003 Pearson Prentice HallDiscrete Random VectorsExam
34、ples: Examples: n nZ=X+Y, Z=X+Y, ( (X X and and Y Y are random execution timesare random execution times) )n nZ = min(X, Y) or Z = max(XZ = min(X, Y) or Z = max(X1 1, X, X2 2,X Xk k) )X X:(X:(X1 1, X, X2 2,X Xk k) ) is a is a k k-dimensional -dimensional rvrv defined on defined on S Snn For each sam
35、ple point For each sample point s s in in S, S,3 - 3 - 4141 2003 Pearson Prentice Hall 2003 Pearson Prentice HallDiscrete Random Vectors (properties) 3 - 3 - 4242 2003 Pearson Prentice Hall 2003 Pearson Prentice HallIndependent Discrete RVsX X andand Y Y are independent are independent iffiff the jo
36、int the joint pmfpmf satisfies:satisfies:Mutual independence also implies:Mutual independence also implies: Pair wise independence vs. set-wide independence Pair wise independence vs. set-wide independence 3 - 3 - 4343 2003 Pearson Prentice Hall 2003 Pearson Prentice HallContinuous Probability Densi
37、ty Function1.1.Mathematical FormulaMathematical Formula2.2.Shows All Values, Shows All Values, x x, & , & Frequencies, f(Frequencies, f(x x) )n nf( f(X X) Is ) Is NotNot Probability Probability3.3.PropertiesProperties(Area Under Curve)(Area Under Curve)ValueValue(Value, Frequency)(Value, Frequency)F
38、requencyFrequencyf(x)f(x)a ab bx xf x dxf x( )()All All X X a x b 10,3 - 3 - 4444 2003 Pearson Prentice Hall 2003 Pearson Prentice HallContinuous Random Variable Probability Probability Is Area Probability Is Area Under Curve!Under Curve! 1984-1994 T/Maker Co.P cxdf x dxc cd d()() f(x)Xcd3 - 3 - 454
39、5 2003 Pearson Prentice Hall 2003 Pearson Prentice HallNormal Distribution3 - 3 - 4646 2003 Pearson Prentice Hall 2003 Pearson Prentice HallImportance of Normal Distribution1. Describes Many Random Processes or Continuous Phenomena2. Can Be Used to Approximate Discrete Probability Distributionsn nEx
40、ample: BinomialExample: Binomial3. Basis for Classical Statistical Inference3 - 3 - 4747 2003 Pearson Prentice Hall 2003 Pearson Prentice HallNormal Distribution1.1.Bell-Shaped & Bell-Shaped & SymmetricalSymmetrical2.2.Mean, Median, Mean, Median, Mode Are EqualMode Are Equal4. 4. Random Variable Ran
41、dom Variable Has Infinite RangeHas Infinite RangeMean Mean Median Median ModeMode3 - 3 - 4848 2003 Pearson Prentice Hall 2003 Pearson Prentice HallProbability Density Functionf f( (x x) )= = Frequency of Random Variable Frequency of Random Variable x x = = Population Standard DeviationPopulation Sta
42、ndard Deviation = = 3.14159; e = 2.718283.14159; e = 2.71828x x= = Value of Random Variable (-Value of Random Variable (- x x ) ) = = Population MeanPopulation Mean3 - 3 - 4949 2003 Pearson Prentice Hall 2003 Pearson Prentice HallEffect of Varying Parameters ( & )3 - 3 - 5050 2003 Pearson Prentice H
43、all 2003 Pearson Prentice HallNormal Distribution ProbabilityProbability is Probability is area under area under curve!curve!3 - 3 - 5151 2003 Pearson Prentice Hall 2003 Pearson Prentice HallInfinite Number of TablesNormal distributions differ by Normal distributions differ by mean & standard deviat
44、ion.mean & standard deviation.3 - 3 - 5252 2003 Pearson Prentice Hall 2003 Pearson Prentice HallInfinite Number of TablesNormal distributions differ by Normal distributions differ by mean & standard deviation.mean & standard deviation.Each distribution would Each distribution would require its own t
45、able.require its own table.Thats an Thats an infinite infinite number!number!3 - 3 - 5353 2003 Pearson Prentice Hall 2003 Pearson Prentice HallNormal Approximation of Binomial DistributionMuMu = = npnpSigma-squared = np(1-p)Sigma-squared = np(1-p)Better approximation with Better approximation with l
46、arger nlarger nn nMore on this when we More on this when we get to the central limit get to the central limit theorem (chapter 6)theorem (chapter 6)n n = 10 = 10 p p = 0.50 = 0.50.0.0.1.1.2.2.3.30 02 24 46 68 81010X XP(X)P(X)3 - 3 - 5454 2003 Pearson Prentice Hall 2003 Pearson Prentice HallInferenti
47、al Statistics3 - 3 - 5555 2003 Pearson Prentice Hall 2003 Pearson Prentice HallStatistical Methods3 - 3 - 5656 2003 Pearson Prentice Hall 2003 Pearson Prentice HallInferential Statistics1.Involves:n nEstimationEstimationn nHypothesis Hypothesis TestingTesting2.Purposen nMake Inferences Make Inferenc
48、es about Population about Population CharacteristicsCharacteristicsPopulation?Population?3 - 3 - 5757 2003 Pearson Prentice Hall 2003 Pearson Prentice HallInference Process3 - 3 - 5858 2003 Pearson Prentice Hall 2003 Pearson Prentice HallInference ProcessPopulation3 - 3 - 5959 2003 Pearson Prentice
49、Hall 2003 Pearson Prentice HallInference ProcessPopulationSample3 - 3 - 6060 2003 Pearson Prentice Hall 2003 Pearson Prentice HallInference ProcessPopulationSampleSample statistic (X)3 - 3 - 6161 2003 Pearson Prentice Hall 2003 Pearson Prentice HallInference ProcessPopulationSampleSample statistic (
50、X)Estimates & tests3 - 3 - 6262 2003 Pearson Prentice Hall 2003 Pearson Prentice Hall1. Random Variables Used to Estimate a Population Parametern nSample Mean, Sample Proportion, Sample Sample Mean, Sample Proportion, Sample MedianMedian2. Example: Sample MeanX Is an Estimator of Population Mean n n
51、If If X X = 3 then = 3 then 3 3 Is the Is the EstimateEstimate of of 3. Theoretical Basis Is Sampling DistributionEstimators3 - 3 - 6363 2003 Pearson Prentice Hall 2003 Pearson Prentice HallSampling Distributions3 - 3 - 6464 2003 Pearson Prentice Hall 2003 Pearson Prentice Hall1. Theoretical Probabi
52、lity Distribution2. Random Variable is Sample Statisticn nSample Mean, Sample Proportion etc.Sample Mean, Sample Proportion etc.3. Results from Drawing All Possible Samples of a Fixed Size4. List of All Possible X, P(X) Pairsn nSampling Distribution of MeanSampling Distribution of MeanSampling Distr
53、ibution3 - 3 - 6565 2003 Pearson Prentice Hall 2003 Pearson Prentice HallExpected Value of X-barRemember “Useful Observation 1”E(X+Y) = E(X) + E(Y)Therefore3 - 3 - 6666 2003 Pearson Prentice Hall 2003 Pearson Prentice HallVariance of X-barRemember Useful Obs. 3 for indep. X, Yn nVar(XVar(X + Y) = +
54、Y) = Var(XVar(X) + ) + VarVar (Y) (Y)ThereforeUseful obs/exercise 4Therefore3 - 3 - 6767 2003 Pearson Prentice Hall 2003 Pearson Prentice HallProperties of Sampling Distribution of Mean3 - 3 - 6868 2003 Pearson Prentice Hall 2003 Pearson Prentice HallProperties of Sampling Distribution of Mean1. Unb
55、iasednessn nMean of Sampling Distribution Equals Population Mean of Sampling Distribution Equals Population MeanMean2. Efficiency (minimum variance)n nSample Mean Comes Closer to Population Mean Sample Mean Comes Closer to Population Mean Than Any Other Unbiased EstimatorThan Any Other Unbiased Esti
56、mator3. Consistencyn nAs Sample Size Increases, Variation of Sample As Sample Size Increases, Variation of Sample Mean from Population Mean DecreasesMean from Population Mean Decreases3 - 3 - 6969 2003 Pearson Prentice Hall 2003 Pearson Prentice HallUnbiasednessUnbiasedUnbiasedBiasedBiased3 - 3 - 70
57、70 2003 Pearson Prentice Hall 2003 Pearson Prentice HallEfficiency Sampling Sampling distribution distribution of medianof medianSampling Sampling distribution distribution of meanof mean3 - 3 - 7171 2003 Pearson Prentice Hall 2003 Pearson Prentice HallConsistencySmaller Smaller sample sample sizesi
58、zeLarger Larger sample sample sizesize 3 - 3 - 7272 2003 Pearson Prentice Hall 2003 Pearson Prentice HallSampling Distribution Solution*Sampling Sampling DistributionDistribution.3830.3830.3830.1915.1915.1915.1915Standardized Standardized Normal DistributionNormal Distribution3 - 3 - 7373 2003 Pears
59、on Prentice Hall 2003 Pearson Prentice HallSampling from Normal Populations3 - 3 - 7474 2003 Pearson Prentice Hall 2003 Pearson Prentice HallSampling from Normal PopulationsCentral TendencyCentral TendencyCentral TendencyDispersionDispersionDispersionSampling Sampling Sampling withwithwith replaceme
60、ntreplacementreplacementPopulation DistributionPopulation DistributionPopulation DistributionSampling DistributionSampling DistributionSampling Distributionn =16n =16 X X = 2.5 = 2.5n = 4n = 4 X X = 5 = 53 - 3 - 7575 2003 Pearson Prentice Hall 2003 Pearson Prentice HallSampling from Non-Normal Popul
61、ations3 - 3 - 7676 2003 Pearson Prentice Hall 2003 Pearson Prentice HallSampling from Non-Normal PopulationsCentral TendencyCentral TendencyCentral TendencyDispersionDispersionDispersionn nnSampling Sampling Sampling withwithwith replacementreplacementreplacementPopulation DistributionPopulation Dis
62、tributionPopulation DistributionSampling DistributionSampling DistributionSampling Distributionn =30n =30 X X = 1.8 = 1.8n = 4n = 4 X X = 5 = 53 - 3 - 7777 2003 Pearson Prentice Hall 2003 Pearson Prentice HallCentral Limit Theorem3 - 3 - 7878 2003 Pearson Prentice Hall 2003 Pearson Prentice HallCent
63、ral Limit Theorem3 - 3 - 7979 2003 Pearson Prentice Hall 2003 Pearson Prentice HallCentral Limit TheoremAs As sample sample size gets size gets large large enough enough (n (n 30) . 30) .3 - 3 - 8080 2003 Pearson Prentice Hall 2003 Pearson Prentice HallCentral Limit TheoremAs As sample sample size g
64、ets size gets large large enough enough (n (n 30) . 30) .sampling sampling distribution distribution becomes becomes almost almost normal.normal.3 - 3 - 8181 2003 Pearson Prentice Hall 2003 Pearson Prentice HallCentral Limit TheoremAs As sample sample size gets size gets large large enough enough (n
65、 (n 30) . 30) .sampling sampling distribution distribution becomes becomes almost almost normal.normal.3 - 3 - 8282 2003 Pearson Prentice Hall 2003 Pearson Prentice HallIntroduction to Estimation3 - 3 - 8383 2003 Pearson Prentice Hall 2003 Pearson Prentice HallStatistical Methods3 - 3 - 8484 2003 Pe
66、arson Prentice Hall 2003 Pearson Prentice HallEstimation Process3 - 3 - 8585 2003 Pearson Prentice Hall 2003 Pearson Prentice HallEstimation ProcessMean, , is unknownPopulationPopulation3 - 3 - 8686 2003 Pearson Prentice Hall 2003 Pearson Prentice HallEstimation ProcessMean, , is unknownPopulationPo
67、pulationRandom SampleRandom SampleMean X = 50Sample3 - 3 - 8787 2003 Pearson Prentice Hall 2003 Pearson Prentice HallEstimation ProcessMean, , is unknownPopulationPopulationRandom SampleRandom SampleI am 95% confident that is between 40 & 60.Mean X = 50Sample3 - 3 - 8888 2003 Pearson Prentice Hall 2
68、003 Pearson Prentice HallUnknown Population Parameters Are Estimated Estimate PopulationParameter.with SampleStatisticMean xProportionp p Variance 2 2 s2 2Differences 1 1 - 2 2 x1 1 - x2 23 - 3 - 8989 2003 Pearson Prentice Hall 2003 Pearson Prentice HallEstimation Methods3 - 3 - 9090 2003 Pearson Pr
69、entice Hall 2003 Pearson Prentice HallEstimation MethodsEstimation3 - 3 - 9191 2003 Pearson Prentice Hall 2003 Pearson Prentice HallEstimation MethodsEstimationPointEstimation3 - 3 - 9292 2003 Pearson Prentice Hall 2003 Pearson Prentice HallEstimation MethodsEstimationPointEstimationIntervalEstimati
70、on3 - 3 - 9393 2003 Pearson Prentice Hall 2003 Pearson Prentice HallPoint Estimation3 - 3 - 9494 2003 Pearson Prentice Hall 2003 Pearson Prentice HallPoint Estimation1. Provides Single Valuen nBased on Observations from 1 SampleBased on Observations from 1 Sample2. Gives No Information about How Clo
71、se Value Is to the Unknown Population Parameter3. Example: Sample MeanX = 3 Is Point Estimate of Unknown Population Mean3 - 3 - 9595 2003 Pearson Prentice Hall 2003 Pearson Prentice HallInterval Estimation3 - 3 - 9696 2003 Pearson Prentice Hall 2003 Pearson Prentice HallEstimation MethodsEstimationP
72、ointEstimationIntervalEstimation3 - 3 - 9797 2003 Pearson Prentice Hall 2003 Pearson Prentice HallInterval Estimation1. Provides Range of Values n nBased on Observations from 1 SampleBased on Observations from 1 Sample2. Gives Information about Closeness to Unknown Population Parametern nStated in t
73、erms of ProbabilityStated in terms of Probability3. Example: Unknown Population Mean Lies Between 50 & 70 with 95% Confidence3 - 3 - 9898 2003 Pearson Prentice Hall 2003 Pearson Prentice HallKey Elements of Interval Estimation3 - 3 - 9999 2003 Pearson Prentice Hall 2003 Pearson Prentice HallKey Elem
74、ents of Interval EstimationSample statistic Sample statistic (point estimate)(point estimate)3 - 3 - 100100 2003 Pearson Prentice Hall 2003 Pearson Prentice HallKey Elements of Interval EstimationConfidence Confidence intervalintervalSample statistic Sample statistic (point estimate)(point estimate)
75、Confidence Confidence limit (lower)limit (lower)Confidence Confidence limit (upper)limit (upper)3 - 3 - 101101 2003 Pearson Prentice Hall 2003 Pearson Prentice HallKey Elements of Interval EstimationConfidence Confidence intervalintervalSample statistic Sample statistic (point estimate)(point estima
76、te)Confidence Confidence limit (lower)limit (lower)Confidence Confidence limit (upper)limit (upper)A A probabilityprobability that the population parameter that the population parameter falls somewhere within the interval.falls somewhere within the interval.3 - 3 - 102102 2003 Pearson Prentice Hall
77、2003 Pearson Prentice HallConfidence Limits for Population MeanWe know the distribution of X-bar (for large n:n n CLT says its normally distributed with CLT says its normally distributed with mean mean MuMu) )For any z, look up Equivalent formulations:3 - 3 - 103103 2003 Pearson Prentice Hall 2003 P
78、earson Prentice HallConfidence Depends on Interval (z)3 - 3 - 104104 2003 Pearson Prentice Hall 2003 Pearson Prentice HallConfidence Depends on Interval (z) x_ X 3 - 3 - 105105 2003 Pearson Prentice Hall 2003 Pearson Prentice HallConfidence Depends on Interval (z) x_ X X = Z x3 - 3 - 106106 2003 Pea
79、rson Prentice Hall 2003 Pearson Prentice HallConfidence Depends on Interval (z)90% Samples90% Samples x_ X X = Z x +1.65+1.65 x x-1.65-1.65 x x3 - 3 - 107107 2003 Pearson Prentice Hall 2003 Pearson Prentice HallConfidence Depends on Interval (z)90% Samples90% Samples95% Samples95% Samples +1.65+1.65
80、 x x x_ X +1.96+1.96 x x-1.65-1.65 x x -1.96-1.96 x x X = Z x3 - 3 - 108108 2003 Pearson Prentice Hall 2003 Pearson Prentice HallConfidence Depends on Interval (z)90% Samples90% Samples95% Samples95% Samples99% Samples99% Samples +1.65+1.65 x x +2.58+2.58 x x x_ X +1.96+1.96 x x -2.58-2.58 x x -1.65
81、-1.65 x x -1.96-1.96 x x X = Z x3 - 3 - 109109 2003 Pearson Prentice Hall 2003 Pearson Prentice Hall1. Probability that the Unknown Population Parameter Falls Within Interval 2. Denoted (1 - n n Is Probability That Parameter Is Is Probability That Parameter Is NotNot Within IntervalWithin Interval3.
82、 Typical Values Are 99%, 95%, 90%Confidence Level 3 - 3 - 110110 2003 Pearson Prentice Hall 2003 Pearson Prentice HallIntervals & Confidence Level Sampling Sampling Distribution Distribution of Meanof MeanIntervals derived from Intervals derived from many samplesmany samplesIntervals Intervals exten
83、d from extend from X - ZX - Z X X to to X + ZX + Z X X (1 - (1 - ) % of ) % of intervals intervals contain contain . . % do not. % do not.3 - 3 - 111111 2003 Pearson Prentice Hall 2003 Pearson Prentice Hall1. Data Dispersionn nMeasured by Measured by 2. Sample Sizen n X X = = / / n n3. Level of Conf
84、idence (1 - )n nAffects ZAffects ZFactors Affecting Interval WidthIntervals Extend fromIntervals Extend from X - ZX - Z X X to to X + ZX + Z X X 1984-1994 T/Maker Co.3 - 3 - 112112 2003 Pearson Prentice Hall 2003 Pearson Prentice HallConfidence Interval Estimates3 - 3 - 113113 2003 Pearson Prentice
85、Hall 2003 Pearson Prentice HallConfidence Interval EstimatesConfidenceIntervals3 - 3 - 114114 2003 Pearson Prentice Hall 2003 Pearson Prentice HallConfidence Interval EstimatesConfidenceIntervalsMean3 - 3 - 115115 2003 Pearson Prentice Hall 2003 Pearson Prentice HallConfidence Interval EstimatesConf
86、idenceIntervalsProportionMean3 - 3 - 116116 2003 Pearson Prentice Hall 2003 Pearson Prentice HallConfidence Interval EstimatesConfidenceIntervalsProportionMeanVariance3 - 3 - 117117 2003 Pearson Prentice Hall 2003 Pearson Prentice HallConfidence Interval EstimatesConfidenceIntervalsProportionMeanVar
87、ianceKnown3 - 3 - 118118 2003 Pearson Prentice Hall 2003 Pearson Prentice HallConfidence Interval EstimatesConfidenceIntervalsProportionMeanVariance UnknownKnown3 - 3 - 119119 2003 Pearson Prentice Hall 2003 Pearson Prentice HallConfidence Interval Estimate Mean ( Known)3 - 3 - 120120 2003 Pearson P
88、rentice Hall 2003 Pearson Prentice HallConfidence Interval EstimatesConfidenceIntervalsProportionMeanVariance UnknownKnown3 - 3 - 121121 2003 Pearson Prentice Hall 2003 Pearson Prentice HallConfidence Interval Mean ( Known)1. Assumptionsn nPopulation Standard Deviation Is KnownPopulation Standard De
89、viation Is Knownn nPopulation Is Normally DistributedPopulation Is Normally Distributedn nIf Not Normal, Can Be Approximated by If Not Normal, Can Be Approximated by Normal Distribution (Normal Distribution (n n 30) 30)3 - 3 - 122122 2003 Pearson Prentice Hall 2003 Pearson Prentice HallConfidence In
90、terval Mean ( Known)ll1.Assumptionsn nPopulation Standard Deviation Is KnownPopulation Standard Deviation Is Knownn nPopulation Is Normally DistributedPopulation Is Normally Distributedn nIf Not Normal, Can Be Approximated by If Not Normal, Can Be Approximated by Normal Distribution (Normal Distribu
91、tion (n n 30) 30)2. Confidence Interval Estimate3 - 3 - 123123 2003 Pearson Prentice Hall 2003 Pearson Prentice HallEstimation Example Mean ( Known)The mean of a random sample of n = 25 isX = 50. Set up a 95% confidence interval estimate for if = 10.3 - 3 - 124124 2003 Pearson Prentice Hall 2003 Pea
92、rson Prentice HallEstimation Example Mean ( Known)The mean of a random sample of n = 25 isX = 50. Set up a 95% confidence interval estimate for if = 10.3 - 3 - 125125 2003 Pearson Prentice Hall 2003 Pearson Prentice HallConfidence Interval Estimate Mean ( Unknown)3 - 3 - 126126 2003 Pearson Prentice
93、 Hall 2003 Pearson Prentice HallConfidence Interval EstimatesConfidenceIntervalsProportionMeanVariance UnknownKnown3 - 3 - 127127 2003 Pearson Prentice Hall 2003 Pearson Prentice HallLarge SamplesThe sample variance s is a good estimator of sigmaCarry on as before3 - 3 - 128128 2003 Pearson Prentice
94、 Hall 2003 Pearson Prentice HallAnother Way To Think About ItDefine variableDefine variableX-bar is the sampling distribution of the mean of a X-bar is the sampling distribution of the mean of a sample of Xssample of XsBy the CLT, X-bar is normally distributedBy the CLT, X-bar is normally distribute
95、dZ is the normalized variable X Z is the normalized variable X n nmumu= 0 and sigma = 1= 0 and sigma = 1Confidence intervalConfidence intervaln nfind z-value associated with desired confidence level find z-value associated with desired confidence level alphaalphan nDe-normalize z-value to compute in
96、terval around X-barDe-normalize z-value to compute interval around X-bar3 - 3 - 129129 2003 Pearson Prentice Hall 2003 Pearson Prentice HallProblem for Small Samples may not be normally distributed is not a good estimator of3 - 3 - 130130 2003 Pearson Prentice Hall 2003 Pearson Prentice HallSolution
97、 for Small Samples1.1.AssumptionsAssumptionsn nPopulation of X Is Population of X Is Normally DistributedNormally Distributed2.Use Students t DistributionUse Students t Distribution1.1.Define variableDefine variable2.2.T has the Student distribution with n-1 degrees of T has the Student distribution
98、 with n-1 degrees of freedom (When X is normally distributed)freedom (When X is normally distributed) Theres a different Student distribution for different Theres a different Student distribution for different degrees of freedom degrees of freedom As n gets large, Student distribution approximates a
99、 As n gets large, Student distribution approximates a normal distribution with mean = 0 and sigma = 1normal distribution with mean = 0 and sigma = 13 - 3 - 131131 2003 Pearson Prentice Hall 2003 Pearson Prentice HallZtStudents t Distribution0t (t (dfdf = 5) = 5)Standard Standard NormalNormalt (t (df
100、df = 13) = 13)Bell-ShapedBell-ShapedSymmetricSymmetricFatter TailsFatter Tails3 - 3 - 132132 2003 Pearson Prentice Hall 2003 Pearson Prentice HallConfidence Interval Mean ( Unknown)Find t-value associated with desired confidence level alpha confidence interval is 3 - 3 - 133133 2003 Pearson Prentice
101、 Hall 2003 Pearson Prentice HallStudents t Table3 - 3 - 134134 2003 Pearson Prentice Hall 2003 Pearson Prentice HallStudents t Table3 - 3 - 135135 2003 Pearson Prentice Hall 2003 Pearson Prentice HallStudents t Tablet valuest values3 - 3 - 136136 2003 Pearson Prentice Hall 2003 Pearson Prentice Hall
102、Students t Tablet valuest values / 2 / 2 / 2 / 23 - 3 - 137137 2003 Pearson Prentice Hall 2003 Pearson Prentice HallStudents t Tablet valuest values / 2 / 2 / 2 / 2Assume:Assume:n n = 3 = 3dfdf = = n n - 1 = 2 - 1 = 2 = .10= .10 /2 =.05/2 =.053 - 3 - 138138 2003 Pearson Prentice Hall 2003 Pearson Pr
103、entice HallStudents t Tablet valuest values / 2 / 2 / 2 / 2Assume:Assume:n n = 3 = 3dfdf = = n n - 1 = 2 - 1 = 2 = .10= .10 /2 =.05/2 =.053 - 3 - 139139 2003 Pearson Prentice Hall 2003 Pearson Prentice HallStudents t Tablet valuest values / 2 / 2Assume:Assume:n n = 3 = 3dfdf = = n n - 1 = 2 - 1 = 2
104、= .10= .10 /2 =.05/2 =.05.05.053 - 3 - 140140 2003 Pearson Prentice Hall 2003 Pearson Prentice HallStudents t TableAssume:Assume:n n = 3 = 3dfdf = = n n - 1 = 2 - 1 = 2 = .10= .10 /2 =.05/2 =.052.9202.920t valuest values / 2 / 2.05.053 - 3 - 141141 2003 Pearson Prentice Hall 2003 Pearson Prentice Ha
105、llDegrees of Freedom (df)1. Number of Observations that Are Free to Vary After Sample Statistic Has Been Calculated2. Examplen nSum of 3 Numbers Is 6Sum of 3 Numbers Is 6X X1 1 = 1 (or Any Number)= 1 (or Any Number)X X2 2 = 2 (or Any Number)= 2 (or Any Number)X X3 3 = = 3 3 (Cannot Vary)(Cannot Vary
106、)Sum = 6Sum = 6degrees of freedom degrees of freedom = = n n -1 -1 = 3 -1= 3 -1= 2= 23 - 3 - 142142 2003 Pearson Prentice Hall 2003 Pearson Prentice HallEstimation Example Mean ( Unknown)A random sample of n = 25 hasx = 50 & s = 8. Set up a 95% confidence interval estimate for .3 - 3 - 143143 2003 P
107、earson Prentice Hall 2003 Pearson Prentice HallEstimation Example Mean ( Unknown)A random sample of n = 25 hasx = 50 & s = 8. Set up a 95% confidence interval estimate for .3 - 3 - 144144 2003 Pearson Prentice Hall 2003 Pearson Prentice HallFinding Sample Sizes3 - 3 - 145145 2003 Pearson Prentice Ha
108、ll 2003 Pearson Prentice HallFinding Sample Sizes for Estimating I dont want to sample too much or too little! Error Is Also Called Bound, Error Is Also Called Bound, B B3 - 3 - 146146 2003 Pearson Prentice Hall 2003 Pearson Prentice HallDetermining Sample SizeZ is determined by desired confidence l
109、evelBut how do you determine sigma?3 - 3 - 147147 2003 Pearson Prentice Hall 2003 Pearson Prentice HallDetermining Sample SizeZ is determined by desired confidence levelBut how do you determine sigma?n nKnown from previous studiesKnown from previous studiesn nPilot test on a small nPilot test on a s
110、mall nn nTheoretical derivationTheoretical derivation3 - 3 - 148148 2003 Pearson Prentice Hall 2003 Pearson Prentice HallSample Size ExampleWhat sample size is needed to be 90% confident of being correct within 5? A pilot study suggested that the standard deviation is 45.3 - 3 - 149149 2003 Pearson
111、Prentice Hall 2003 Pearson Prentice HallSample Size ExampleWhat sample size is needed to be 90% confident of being correct within 5? A pilot study suggested that the standard deviation is 45.3 - 3 - 150150 2003 Pearson Prentice Hall 2003 Pearson Prentice HallEnd of ChapterAny blank slides that follo
112、w are blank intentionally.3 - 3 - 151151 2003 Pearson Prentice Hall 2003 Pearson Prentice HallIndependent Discrete RVsX X andand Y Y are independent are independent iffiff the joint the joint pmfpmf satisfies:satisfies:Mutual independence also implies:Mutual independence also implies: Pair wise inde
113、pendence vs. set-wide independence Pair wise independence vs. set-wide independence 3 - 3 - 152152 2003 Pearson Prentice Hall 2003 Pearson Prentice HallDiscrete Convolution Let Let Z=X+Y . Z=X+Y . Then, if Then, if X X and and Y Y are independent,are independent, In general, then,In general, then,3
114、- 3 - 153153 2003 Pearson Prentice Hall 2003 Pearson Prentice HallDefinitionsDistribution function: Distribution function: If If F FX X(x) (x) is a continuous function of is a continuous function of x, x, then then X X is a is a continuous random variable.continuous random variable.n nF FX X(x): (x)
115、: discrete indiscrete in x x Discrete Discrete rvsrvsn nF FX X(x): (x): piecewise continuouspiecewise continuous Mixed Mixed rvsrvs 3 - 3 - 154154 2003 Pearson Prentice Hall 2003 Pearson Prentice HallDefinitions (Continued)(Continued)Equivalence:CDF (cumulative distribution function)PDF (probability
116、 distribution function) Distribution functionFX(x) or FX(t) or F(t) 3 - 3 - 155155 2003 Pearson Prentice Hall 2003 Pearson Prentice HallProbability Density Function (pdf)X X : continuous : continuous rvrv, then, then, pdfpdf properties:properties:1.1. 2.2. 3 - 3 - 156156 2003 Pearson Prentice Hall 2
117、003 Pearson Prentice HallDefinitions(Continued)(Continued)Equivalence: pdf n nprobability density functionprobability density functionn ndensity functiondensity functionn ndensitydensityn nf(t) = f(t) = For a non-negativerandom variable3 - 3 - 157157 2003 Pearson Prentice Hall 2003 Pearson Prentice
118、HallExponential DistributionArises commonly in reliability & queuing theory.Arises commonly in reliability & queuing theory.A non-negative random variableA non-negative random variableIt exhibits It exhibits memorylessmemoryless (Markov) property. (Markov) property.Related to (the discrete) Poisson
119、distribution Related to (the discrete) Poisson distribution n nInterarrival time between two IP packets (or voice Interarrival time between two IP packets (or voice calls)calls)n nTime to failure, time to repair etc.Time to failure, time to repair etc. Mathematically (CDF and Mathematically (CDF and
120、 pdfpdf, respectively): , respectively): 3 - 3 - 158158 2003 Pearson Prentice Hall 2003 Pearson Prentice Hall CDF of exponentially distributed random variable with = 0.0001tF(t)12500 25000 37500 500003 - 3 - 159159 2003 Pearson Prentice Hall 2003 Pearson Prentice HallExponential Density Function (pdf)f(t)t
