Fundamentals of Probability and Statistics for Engineers

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Parameter Estimation 305g(p)L 1L 21pFigure 9.11 Parabola defined by Equation (9.150)Upon transposing the right-hand side, we have!!p 2 1 ‡ u2 =2np 2X ‡ u2 =2n‡ X 2 0: …9:150†In Equation (9.150), the left-hand side defines a parabola, as shown in Figure9.11, and two roots L 1 and L 2 of Equation (9.150) with the equal sign define theinterval within which the parabola is negative. Hence, solving the quadraticequation defined by Equation (9.150), we have8 !
306 Fundamentals of Probability and Statistics for EngineersIn this approximation, sample mean X is at the center of the interval forwhich the width is a function of the sample and the sample size.Example 9.21. Problem: in a random sample of 500 persons in the city of LosAngeles it was found that 372 did not approve of US energy policy. Determinea 95% confidence interval for p, the actual proportion of the Los Angelespopulation registering disapproval.Answer: in this example, n ˆ 500, ˆ 0.05, and the observed sample mean isx ˆ 372/500 ˆ 0.74. Table A.3 gives u 0:025 ˆ 1:96. Substituting these valuesinto Equation (9.153) then yieldsP…0:740:04 and Bancroft, T.A., 1952, Statistical Theory in Research, McGraw-Hill,New York.Benedict, T.R., and Soong, T.T., 1967, ‘‘The Joint Estimation of Signal and Noise fromthe Sum Envelope’’, IEEE Trans. Information Theory, IT-13 447–454.Fisher, R.A., 1922, ‘‘On the Mathematical Foundation of Theoretical Statistics’’, Phil.Trans. Roy. Soc. London, Ser. A 222 309–368.Gerlough, D.L., 1955, ‘‘The Use of Poisson Distribution in Traffic’’, in Poisson andTraffic, The Eno Foundation for Highway Traffic Control, Saugatuk, CT.Mood, A.M., 1950, Introduction to the Theory of Statistics, McGraw-Hill, New York.Neyman, J., 1935, ‘‘Su un Teorema Concernente le Cosiddeti Statistiche Sufficienti’’,Giorn. Inst. Ital. Atturi. 6 320–334.Pearson, K., 1894, ‘‘Contributions to the Mathematical Theory of Evolution’’, Phil.Trans. Roy. Soc. London, Ser. A 185 71–78.Soong, T.T., 1969, ‘‘An Extension of the Moment Method in Statistical Estimation’’,SIAM J. Appl. Math. 17 560–568.Wilks, S.S., 1962, Mathematical Statistics, John Wiley & Sons Inc., New York.FURTHER READING AND COMMENTSThe Cramér–Rao inequality is named after two well-known statisticians, H. Crame´r andC.R. Rao, who independently established this result in the following references. However,this inequality was first stated by Fisher in 1922 (see the Reference section). In fact,much of the foundation of parameter estimation and statistical inference in general, suchas concepts of consistency, efficiency, and sufficiency, was laid down by Fisher in a seriesof publications, given below.Cramér, H., 1946, Mathematical Methods of Statistics, Princeton University Press,Princeton, NJ.Fisher, R.A., 1924, ‘‘On a Distribution Yielding the Error Functions of Several WellknownStatistics’’, Proc. Int. Math. Congress, Vol. II, Toronto, 805–813.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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