Fundamentals of Probability and Statistics for Engineers

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Model Verification 317alternative that the probability distribution of X is not of the stated type on thebasis of a sample of size n from population X. One of the most popular and mostversatile tests devised for this purpose is the chi-squared ( 2 )goodness-of-fittest introduced by Pearson (1900).10.2.1 THE CASE OF KNOWN PARAMETERSLet us first assume that the hypothesized distribution is completely specifiedwith no unknown parameters. In order to test hypothesis H, some statistich(X 1 ,X 2 ,...,X n ) of the sample is required that gives a measure of deviation ofthe observed distribution as constructed from the sample from the hypothesizeddistribution.In the 2 test, the statistic used is related to, roughly speaking, the differencebetween the frequency diagram constructed from the sample and a correspondingdiagram constructed from the hypothesized distribution. Let the rangespace of X be divided into k mutually exclusive intervals A 1 ,A 2 ,..., and A k ,and let N i be the number of X j falling into A i , i ˆ 1, 2, . . . , k. Then, the observedprobabilities P(A i ) are given byobserved P…A i †ˆNi; i ˆ 1; 2; ...; k: …10:1†nThe theoretical probabilities P(A i ) can be obtained from the hypothesizedpopulation distribution. Let us denote these bytheoretical P…A i †ˆp i ; i ˆ 1; 2; ...; k: …10:2†A logical choice of a statistic giving a measure of deviation isX kc iiˆ1 N 2 ip in; …10:3†which is a natural least-square type deviation measure. Pearson (1900) showedthat, if we take coefficient c i ˆ n/p i , the statistic defined by Expression (10.3)has particularly simple properties. Hence, we choose as our deviation measureD ˆ Xkˆ Xknpiˆ1 iiˆ1N inN 2 inp in:p i 2ˆ Xkiˆ1…N i np i † 2np i…10:4†TLFeBOOK
318 Fundamentals of Probability and Statistics for EngineersLet us note that D is a statistic since it is a function of N i , which are, in turn,functions of sample X 1 ,...,X n . The distribution of statistic D is given inTheorem 10.1, attributable to Pearson (1900).Theorem 10. 1: assuming that hypothesis H is true, the distribution of Ddefined by Equation (10.4) approaches a chi-squared distribution with (k 1)degrees of freedom as n !1. Its pdf is given by [see Equation (7.67)]8 h i
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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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