Basic Mathematical Statistics

 Basic Mathematical Statistics Mathematical statistics deals basically with situations which can be described as follows: Given a population of statistically identical, independent elements with unknown (statistical) properties, measurements regarding these properties are made on a (random) sample of this population and, on the basis of the collected data, conclusions are made for the remaining elements of the population. Examples are the parameter estimation for the distribution function of an item's failure-free operating time 't, or the decision whether the expected value (mean) of't is greater than a given value. Mathematical statistics thus goes from observations (realizations) of a given (random) event in a series of independent trials and searches for a suitable probabilistic model for the event considered (inductive approach). Methods used are based on probability theory and results obtained can only be formulated in a probabilistic language. Minimization of the risk for a false conclusion is an important objective in mathematical statistics. This Appendix introduces the basic concepts of mathematical statistics used in planning and evaluating quality and reliability tests, as given in Chapter 7. Emphased are empirical methods, (statistical) parameter estimation, and (statistical) testing of hypotheses. To simplify the notation, the terms random and statistical (in brackets) will often be omitted, and mean stands for expected value. This appendix is a compendium of mathematical statistics, consistent from a mathematical point of view but still with engineering applications in mind. AS.I Empirical Methods Empirical methods allow a quick and easy estimation of the distribution function as well as of the mean, variance, and other moments characterizing a random variable. These estimates are generally based on the empirical distribution function and have thus great intuitive appeal. A. Birolini, Reliability Engineering © Springer-Verlag Berlin Heidelberg 1999 436 AS Basic Mathematical Statistics AS.I.I Empirical Distribution Function A sample of size n of a random variable 't with the distribution function F(t) is a random vector t = ('tl' ... , 'tn) whose components 'ti are tacitly assumed to be independent and identically distributed random variables with F(t) = Pr{'ti ::; t}, i = 1, ... , n. For example, 'tl> ... , 'tn can be the failure-free operating times of n items randomly selected from a lot of statistically identical items, with a distribution function F(t) of the failure-free operating time 't. The observed failure-free operating times, i.e. the realization of the random vector t = ('tl' ... , 'tn), is a set tl> ... , tn of positive real values. The distinction between random variables 'tl, ... , 'tn and their observations tl> ... , tn is important from a mathematical point of view:) When the sample elements are ordered by increasing magnitude, an ordered sample t(l)' ... , t(n) is obtained. The corresponding ordered observations are t(l)' ... , t(n)' For a set of ordered observations t(1)'"'' t(n)' the right continuous function j 0 ~ i F (t) = - n n I for I < 1(1) (AS.I) for I ~ I(n) is the empirical distribution function of the random variable 't, see Fig. AS.l for a graphical representation. Fn(t) expresses the relative frequency of the event {'t::; t} in n independent trial repetitions and provides a well defined estimate of the distribution function F(t) = Pr{'t::; t}. In the following, the symbol ~ is used to denote an estimate of an unknown quantity. As mentioned in the footnote below, when investigating the properties of the empirical distribution function Fn(t) it is necessary in Eq. (AS.I) to replace the observat!ons t(1)' ... , t(n) by the sample elements 't(1)' ... , 't(n)' For any given value of t, nFn(t) is a binomially-distributed random variable (Eq. (A6.120)) with parameter p = F(t). Thus Fn(t) has mean (AS.2) and variance *) The investigation of statistical methods and the discussion of their properties can only be base

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