Page 264 - Fundamentals of Probability and Statistics for Engineers

8

Observed Data and Graphical

Representation

Referring to Figure 1.1 in Chapter 1, we are concerned in this and subsequent
chapters with step D ! E of the basic cycle in probabilistic modeling, that is,
parameter estimation and model verification on the basis of observed data. In
Chapters 6 and 7, our major concern has been the selection of an appropriate
model (probability distribution) to represent a physical or natural phenom-
enon based on our understanding of its underlying properties. In order to
specify the model completely, however, it is required that the parameters in the
distribution be assigned. We now consider this problem of parameter estima-
tion using available data. Included in this discussion are techniques for asses-
sing the reasonableness of a selected model and the problem of selecting a
model from among a number of contending distributions when no single one
is preferred on the basis of the underlying physical characteristics of a given
phenomenon.
Let us emphasize at the outset that, owing to the probabilistic nature of the
situation, the problem of parameter estimation is precisely that – an estima-
tion problem. A sequence of observations, say n in number, is a sample of
observed values of the underlying random variable. If we were to repeat the
sequence of n observations, the random nature of the experiment should
produce a different sample of observed values. Any reasonable rule for
extracting parameter estimates from a set of n observations will thus give
different estimates for different sets of observations. In other words, no single
sequence of observations, finite in number, can be expected to yield true
parameter values. What we are basically interested in, therefore, is to obtain
relevant information about the distribution parameters by actually observing
the underlying random phenomenon and using these observed numerical
values in a systematic way.

Fundamentals of Probability and Statistics for Engineers T.T. Soong  2004 John Wiley & Sons, Ltd
ISBNs: 0-470-86813-9 (HB) 0-470-86814-7 (PB)

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