ChapProbabilityFacultyofBiomedicalEngineeringIran概率生物医学工程学院伊朗实用PPT课件

1、Experiments, Outcomes, & Events第1页/共159页Experiments & Outcomes 1.Experiment Process of Obtaining an Observation, Outcome or Simple Event 2.Sample Space (S) Collection of All Possible Outcomes第2页/共159页Outcome ExamplesToss a Coin, Note FaceHead, TailToss 2 Coins, Note FacesHH, HT, TH, TTPlay a

2、 Football GameWin, Lose, TieInspect a Part, Note QualityDefective, OKObserve GenderMale, Female第3页/共159页Events Any Collection of Sample Points (outcomes) Simple Event Collection of outcomes thats simple to describe Compound Event Collection of outcomes that is described as unions or intersections of

3、 other events第4页/共159页Event ExamplesSample SpaceHH, HT, TH, TT1 Head & 1 TailHT, THHeads on 1st CoinHH, HTAt Least 1 HeadHH, HT, THHeads on BothHH第5页/共159页Sample Space第6页/共159页Visualizing Sample Space1.Listing S = Head, Tail2.Venn Diagram 3.Contingency Table4.Decision Tree Diagram第7页/共159页Venn D

4、iagram第8页/共159页Contingency Table第9页/共159页Tree Diagram第10页/共159页Probabilities第11页/共159页What is Probability?1. Numerical Measure of Likelihood that Event Will OccurP(Event)P(A)Prob(A)2. Lies Between 0 & 13. Sum of outcome probabilities is 1第12页/共159页Probability P(A)=limn(A)/N0N0第13页/共159页Many Repe

5、titions!第14页/共159页Conditional Probability第15页/共159页Conditional Probability1. Event Probability Given that Another Event Occurred2. Revise Original Sample Space to Account for New InformationEliminates Certain Outcomes3.P(A | B) = P(A and B) P(B)第16页/共159页Conditional Probability Using Venn Diagram第17

6、页/共159页Conditional Probability Using Contingency TableP(Ace | Black) = P(Ace AND Black)P(Black)2 5226 52226/第18页/共159页Statistical Independence 1.Event Occurrence Does Not Affect Probability of Another Event P(A | B) = P(A) Example: Toss 1 Coin Twice (independent) P(second toss H)= P(second toss H |

7、first toss H) = 第19页/共159页Tree Diagram第20页/共159页Thinking ChallengeUsing the Table Then the Formula, Whats the Probability?Pr(C)=P(B|C) =P(C|B) = Are C & B Independent?第21页/共159页Solution*Using the Formula, the Probabilities Are:第22页/共159页Multiplicative Rule第23页/共159页Multiplicative Rule 1.Used to

8、Get Compound Probabilities for Intersection of Events Called Joint Events 2.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)第24页/共159页Multiplicative Rule Example第25页/共159页Thinking ChallengeUsing the Multiplicative Rule, Whats the Probability?P(

9、C B) =P(B D) =P(A B) =第26页/共159页Solution*Using the Multiplicative Rule, the Probabilities Are:第27页/共159页Independence Revisited If A is independent of B, B is independent of A P(A and B) = P(B|A)P(A)=P(A|B)P(B) P(A|B)=P(A) P(B|A)P(A) = P(A)P(B) P(B|A)=P(B) Equivalence of the two independence definiti

10、ons: P(A and B) = P(A)*P(B) if and only if P(B|A) = P(B) P(A and B) = P(A)P(B|A) If 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)第28页/共159页Random Variable第29页/共159页Random VariablesA random variable (rv) X is a mapping (function) from the sample space S to t

11、he set of real numbers If image(X ) finite or countable infinite, X is a discrete rvInverse image of a real number x is the set of all sample points that are mapped by X into x:It is easy to see that 第30页/共159页Discrete Random Variable: pmfpk第31页/共159页Discrete Random Variable: CDF第32页/共159页Probabilit

12、y Mass Function (pmf)Ax : set of all sample points such that, pmf 第33页/共159页pmf Properties Since a discrete rv X takes a finite or a countably infinite set values, the last property above can be restated as,第34页/共159页Distribution Functionpmf: defined for a specific rv value, i.e.,Probability of a se

13、t Cumulative Distribution Function (CDF) 第35页/共159页Distribution Function properties 第36页/共159页Equivalence: Probability mass function Discrete density function(consider integer valued random variable)cdf:pmf: )(kXPpk xkkpxF0)(Discrete Random Variables) 1()(kFkFpk第37页/共159页Common discrete random varia

14、bles Constant Uniform Bernoulli Binomial Geometric Poisson Exponential第38页/共159页Discrete Random VectorsExamples: Z=X+Y, (X and Y are random execution times) Z = min(X, Y) or Z = max(X1, X2,Xk)X:(X1, X2,Xk) is a k-dimensional rv defined on S For each sample point s in S,第39页/共159页Discrete Random Vect

15、ors (properties) 第40页/共159页Independent Discrete RVs X and Y are independent iff the joint pmf satisfies: Mutual independence also implies: Pair wise independence vs. set-wide independence 第41页/共159页Continuous Probability Density Function1.Mathematical Formula2.Shows All Values, x, & Frequencies,

16、 f(x) f(X) Is Not Probability3.Properties 第42页/共159页Continuous Random Variable Probability 1984-1994 T/Maker Co. Xcd第43页/共159页Normal Distribution第44页/共159页Importance of Normal Distribution 1.Describes Many Random Processes or Continuous Phenomena 2.Can Be Used to Approximate Discrete Probability Dis

17、tributions Example: Binomial 3.Basis for Classical Statistical Inference第45页/共159页Normal Distribution1.Bell-Shaped & Symmetrical2.Mean, Median, Mode Are Equal4. Random Variable Has Infinite RangeXf(X)第46页/共159页Probability Density Functionf(x)= Frequency of Random Variable x= Population Standard

18、Deviation = 3.14159; e = 2.71828x= Value of Random Variable (- x )= Population Mean第47页/共159页Effect of Varying Parameters ( & )Xf(X)CAB第48页/共159页Normal Distribution Probability?)()(dxxfdxcPdccdxf(x)第49页/共159页Xf(X)Infinite Number of Tables第50页/共159页Xf(X)Infinite Number of Tables第51页/共159页Normal A

19、pproximation of Binomial DistributionMu = npSigma-squared = np(1-p)Better approximation with larger n More on this when we get to the central limit theorem (chapter 6)第52页/共159页Inferential Statistics第53页/共159页Statistical MethodsStatisticalMethodsDescriptiveStatisticsInferentialStatistics第54页/共159页In

20、ferential Statistics1.Involves: Estimation Hypothesis Testing2.Purpose Make Inferences about Population Characteristics第55页/共159页Inference Process第56页/共159页Inference Process第57页/共159页Inference Process第58页/共159页Inference Process第59页/共159页Inference Process第60页/共159页 1.Random Variables Used to Estimate

21、 a Population Parameter Sample Mean, Sample Proportion, Sample Median 2.Example: Sample MeanX Is an Estimator of Population Mean IfX = 3 then 3 Is the Estimate of 3.Theoretical Basis Is Sampling DistributionEstimators第61页/共159页Sampling Distributions第62页/共159页 1.Theoretical Probability Distribution 2

22、.Random Variable is Sample Statistic Sample Mean, Sample Proportion etc. 3.Results from Drawing All Possible Samples of a Fixed Size 4.List of All Possible X, P(X) Pairs Sampling Distribution of MeanSampling Distribution第63页/共159页Expected Value of X-bar Remember “Useful Observation 1”E(X+Y) = E(X) +

23、 E(Y) Therefore111iiiXE XEEXnnE Xnnn第64页/共159页Variance of X-bar Remember Useful Obs. 3 for indep. X, Y Var(X + Y) = Var(X) + Var (Y) Therefore Useful obs/exercise 4 ThereforeiiVarXVar X2Var kXk Var X()2211iiiiXVar XVarVar XnnVar XnVar Xnn1XXn第65页/共159页Properties of Sampling Distribution of Mean第66页/

24、共159页Properties of Sampling Distribution of Mean 1.Unbiasedness Mean of Sampling Distribution Equals Population Mean 2.Efficiency (minimum variance) Sample Mean Comes Closer to Population Mean Than Any Other Unbiased Estimator 3.Consistency As Sample Size Increases, Variation of Sample Mean from Pop

25、ulation Mean Decreases第67页/共159页Unbiasedness XP( X)CA第68页/共159页Efficiency XP( X)AB第69页/共159页Consistency XP( X)AB第70页/共159页Sampling Distribution Solution*8 X = .47.88.2 XSampling DistributionStandardized Normal Distribution第71页/共159页Sampling from Normal Populations第72页/共159页Sampling from Normal Popul

26、ationsCentral TendencyDispersionSampling with replacement第73页/共159页Sampling from Non-Normal Populations第74页/共159页Sampling from Non-Normal PopulationsCentral TendencyDispersion Sampling with replacement第75页/共159页Central Limit Theorem第76页/共159页Central Limit Theorem第77页/共159页XCentral Limit Theorem第78页/

27、共159页XCentral Limit Theorem第79页/共159页XCentral Limit Theorem第80页/共159页Introduction to Estimation第81页/共159页Statistical MethodsStatisticalMethodsDescriptiveStatisticsInferentialStatisticsEstimationHypothesisTesting第82页/共159页Estimation Process第83页/共159页Estimation ProcessMean, , is unknown第84页/共159页Estim

28、ation ProcessMean, , is unknownMean X = 50Sample第85页/共159页Estimation ProcessMean, , is unknownI am 95% confident that is between 40 & 60.Mean X = 50Sample第86页/共159页Unknown Population Parameters Are Estimated 第87页/共159页Estimation Methods第88页/共159页Estimation MethodsEstimation第89页/共159页Estimation M

29、ethodsEstimationPointEstimation第90页/共159页Estimation MethodsEstimationPointEstimationIntervalEstimation第91页/共159页Point Estimation第92页/共159页Point Estimation 1.Provides Single Value Based on Observations from 1 Sample 2.Gives No Information about How Close Value Is to the Unknown Population Parameter 3

30、.Example: Sample MeanX = 3 Is Point Estimate of Unknown Population Mean第93页/共159页Interval Estimation第94页/共159页Estimation MethodsEstimationPointEstimationIntervalEstimation第95页/共159页Interval Estimation 1.Provides Range of Values Based on Observations from 1 Sample 2.Gives Information about Closeness

31、to Unknown Population Parameter Stated in terms of Probability 3.Example: Unknown Population Mean Lies Between 50 & 70 with 95% Confidence第96页/共159页Key Elements of Interval Estimation第97页/共159页Key Elements of Interval Estimation第98页/共159页Key Elements of Interval Estimation第99页/共159页Key Elements

32、of Interval Estimation第100页/共159页Confidence Limits for Population Mean We know the distribution of X-bar (for large n: CLT says its normally distributed with mean Mu) For any z, look up Equivalent formulations:XXzPrXXz(,XXXzXz,XXXzzXzXz第101页/共159页Confidence Depends on Interval (z)第102页/共159页Confiden

33、ce Depends on Interval (z)第103页/共159页Confidence Depends on Interval (z)第104页/共159页Confidence Depends on Interval (z)第105页/共159页Confidence Depends on Interval (z)第106页/共159页Confidence Depends on Interval (z)第107页/共159页 1.Probability that the Unknown Population Parameter Falls Within Interval 2.Denote

34、d (1 - Is Probability That Parameter Is Not Within Interval 3.Typical Values Are 99%, 95%, 90%Confidence Level 第108页/共159页Intervals & Confidence Level x = 1 - /2 /2X_ x_第109页/共159页 1.Data Dispersion Measured by 2.Sample SizeX = / n 3.Level of Confidence (1 - ) Affects ZFactors Affecting Interval

35、 Width 1984-1994 T/Maker Co.第110页/共159页Confidence Interval Estimates第111页/共159页Confidence Interval EstimatesConfidenceIntervals第112页/共159页Confidence Interval EstimatesConfidenceIntervalsMean第113页/共159页Confidence Interval EstimatesConfidenceIntervalsProportionMean第114页/共159页Confidence Interval Estima

36、tesConfidenceIntervalsProportionMeanVariance第115页/共159页Confidence Interval EstimatesConfidenceIntervalsProportionMeanVariance Known第116页/共159页Confidence Interval EstimatesConfidenceIntervalsProportionMeanVariance Unknown Known第117页/共159页Confidence Interval Estimate Mean ( Known)第118页/共159页Confidence

37、 Interval EstimatesConfidenceIntervalsProportionMeanVariance Unknown Known第119页/共159页Confidence Interval Mean ( Known) 1.Assumptions Population Standard Deviation Is Known Population Is Normally Distributed If Not Normal, Can Be Approximated by Normal Distribution (n 30)第120页/共159页Confidence Interva

38、l Mean ( Known)l1.Assumptions Population Standard Deviation Is Known Population Is Normally Distributed If Not Normal, Can Be Approximated by Normal Distribution (n 30) 2.Confidence Interval Estimate第121页/共159页Estimation Example Mean ( Known)The mean of a random sample of n = 25 isX = 50. Set up a 9

39、5% confidence interval estimate for if = 10.第122页/共159页Estimation Example Mean ( Known)The mean of a random sample of n = 25 isX = 50. Set up a 95% confidence interval estimate for if = 10.第123页/共159页Confidence Interval Estimate Mean ( Unknown)第124页/共159页Confidence Interval EstimatesConfidenceInterv

40、alsProportionMeanVariance Unknown Known第125页/共159页Large Samples The sample variance s is a good estimator of sigma Carry on as before第126页/共159页Another Way To Think About ItDefine variableX-bar is the sampling distribution of the mean of a sample of XsBy the CLT, X-bar is normally distributedZ is th

41、e normalized variable X mu= 0 and sigma = 1Confidence intervalfind z-value associated with desired confidence level alphaDe-normalize z-value to compute interval around X-bar/XXXXZnsn第127页/共159页Problem for Small Samples may not be normally distributed is not a good estimator ofXsnX第128页/共159页Solutio

42、n for Small Samples 1.Assumptions Population of X Is Normally Distributed2.Use Students t Distribution1.Define variable2.T has the Student distribution with n-1 degrees of freedom (When X is normally distributed) Theres a different Student distribution for different degrees of freedom As n gets larg

43、e, Student distribution approximates a normal distribution with mean = 0 and sigma = 1/XTsn第129页/共159页Students t Distribution第130页/共159页Confidence Interval Mean ( Unknown) Find t-value associated with desired confidence level alpha confidence interval is /XTsn21/ ,PrnTt2121/ ,/ ,nnssXtXtnn100 1第131页

44、/共159页Students t Table第132页/共159页vt.10t.05t.02513.078 6.31412.70621.886 2.920 4.30331.638 2.353 3.182Students t Table第133页/共159页Students t Tablevt.10t.05t.02513.078 6.31412.70621.886 2.920 4.30331.638 2.353 3.182第134页/共159页Students t Tablevt.10t.05t.02513.078 6.31412.70621.886 2.920 4.30331.638 2.35

45、3 3.182第135页/共159页Students t Tablevt.10t.05t.02513.078 6.31412.70621.886 2.920 4.30331.638 2.353 3.182第136页/共159页vt.10t.05t.02513.078 6.31412.70621.886 2.920 4.30331.638 2.353 3.182Students t Table第137页/共159页vt.10t.05t.02513.078 6.31412.70621.886 2.920 4.30331.638 2.353 3.182Students t Table第138页/共1

46、59页vt.10t.05t.02513.078 6.31412.70621.886 2.920 4.30331.638 2.353 3.182Students t Table第139页/共159页Degrees of Freedom (df) 1.Number of Observations that Are Free to Vary After Sample Statistic Has Been Calculated 2.Example Sum of 3 Numbers Is 6X1 = 1 (or Any Number)X2 = 2 (or Any Number)X3 = 3 (Canno

47、t Vary)Sum = 6第140页/共159页Estimation Example Mean ( Unknown)A random sample of n = 25 hasx = 50 & s = 8. Set up a 95% confidence interval estimate for .第141页/共159页Estimation Example Mean ( Unknown)A random sample of n = 25 hasx = 50 & s = 8. Set up a 95% confidence interval estimate for .第142

48、页/共159页Finding Sample Sizes第143页/共159页Finding Sample Sizes for Estimating I dont want to sample too much or too little! 第144页/共159页Determining Sample Size Z is determined by desired confidence level But how do you determine sigma?第145页/共159页Determining Sample Size Z is determined by desired confiden

49、ce level But how do you determine sigma? Known from previous studies Pilot test on a small n Theoretical derivation第146页/共159页Sample 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.第147页/共159页Sample Size ExampleWha