Fundamentals of Probability and Statistics for Engineers

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8Observed Data and GraphicalRepresentationReferring to Figure 1.1 in Chapter 1, we are concerned in this and subsequentchapters with step D ! E of the basic cycle in probabilistic modeling, that is,parameter estimation and model verification on the basis of observed data. InChapters 6 and 7, our major concern has been the selection of an appropriatemodel (probability distribution) to represent a physical or natural phenomenonbased on our understanding of its underlying properties. In order tospecify the model completely, however, it is required that the parameters in thedistribution be assigned. We now consider this problem of parameter estimationusing available data. Included in this discussion are techniques for assessingthe reasonableness of a selected model and the problem of selecting amodel from among a number of contending distributions when no single oneis preferred on the basis of the underlying physical characteristics of a givenphenomenon.Let us emphasize at the outset that, owing to the probabilistic nature of thesituation, the problem of parameter estimation is precisely that – an estimationproblem. A sequence of observations, say n in number, is a sample ofobserved values of the underlying random variable. If we were to repeat thesequence of n observations, the random nature of the experiment shouldproduce a different sample of observed values. Any reasonable rule forextracting parameter estimates from a set of n observations will thus givedifferent estimates for different sets of observations. In other words, no singlesequence of observations, finite in number, can be expected to yield trueparameter values. What we are basically interested in, therefore, is to obtainrelevant information about the distribution parameters by actually observingthe underlying random phenomenon and using these observed numericalvalues in a systematic way.Fundamentals of Probability and Statistics for Engineers T.T. Soong © 2004 John Wiley & Sons, LtdISBNs: 0-470-86813-9 (HB) 0-470-86814-7 (PB)TLFeBOOK
248 Fundamentals of Probability and Statistics for Engineers8.1 HISTOGRAM AND FREQUENCY DIAGRAMSGiven a set of independent observations x 1 ,x 2 ,..., and x n of a random variableX, a useful first step is to organize and present them properly so that they canbe easily interpreted and evaluated. When there are a large number of observeddata, a histogram is an excellent graphical representation of the data, facilitating(a)anevaluationofadequacyoftheassumedmodel,(b)estimationofpercentilesof the distribution, and (c) estimation of the distribution parameters.Let us consider, for example, a chemical process that is producing batches ofa desired material; 200 observed values of the percentage yield, X, representinga relatively large sample size, are given in Table 8.1 (Hill, 1975). The samplevalues vary from 64 to 76. Dividing this range into 12 equal intervals andplotting the total number of observed yields in each interval as the height ofa rectangle over the interval results in the histogram as shown in Figure 8.1.A frequency diagram is obtained if the ordinate of the histogram is divided bythe total number of observations, 200 in this case, and by the interval width D(which happens to be one in this example). We see that the histogram orthe frequency diagram gives an immediate impression of the range, relativefrequency, and scatter associated with the observed data.In thecaseof a discreterandom variable, thehistogram and frequency diagram asobtained from observed data take the shape of a bar chart as opposed to connectedrectangles in the continuous case. Consider, for example, the distribution of thenumber of accidents per driver during a six-year time span in California. The data500.25Histogram403020N(70,4)0.200.150.10Frequency diagram100.0564 65 66 67 68 69 70 71 72 73 74 75 76Percentage yieldFigure 8.1Histogram and frequency diagram for percentage yield(data source: Hill, 1975)TLFeBOOK
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ContentsPREFACExiii1 INTRODUCTION 1
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Contentsix6.1.2 Geometric Distribut
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ContentsxiAPPENDIX A: TABLES 365A.1
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PrefaceThis book was written for an
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1IntroductionAt present, almost all
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Introduction 3may be rejected at th
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Part AProbability and Random Variab
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390 Subject IndexExtreme-value dist
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