Page 15 - Probability and Statistical Inference

viii Preface

the collection. With the help of interesting examples and discussions, Section
3.7 briefly unfolds the intricate relationship between “zero correlation” and
“independence” for two random variables.
b) In Chapter 4, the Helmert transformation for a normal distribution,
and the transformation involving the spacings for an exponential distribution,
have both been developed thoroughly. The related remarks are expected to
make many readers pause and think. Section 4.6 exposes readers to some
continuous multivariate distributions other than the multivariate normal.
Section 4.7 has special messages – in defining a random variable having
the Student’s t or F distribution, for example, one takes independent random
variables in the numerator and denominator. But, what happens when the
random variables in the numerator and denominator are dependent? Some
possible answers are emphasized with the help of examples. Exercise 4.7.4
shows a way to construct examples where the distribution of a sample
variance is a multiple of Chi-square even though the random samples do
not come from a normal population!
c) The derivation of the central limit theorem for the sample variance
(Theorem 5.3.6) makes clever use of several non-trivial ingredients from
the theory of probability. In other words, this result reinforces the importance
of many results taught in the preceding sections. That should be an important
aspect of learning. No book at this level highlights this in the way I have. In
Section 5.4, various convergence properties of the densities and percentage
points of the Student’s t and F distributions, for example, are laid out. The
usefulness of such approximations is emphasized through computation. In
no other book like this will one find such engaging discussions and
comparisons.
d) No book covers the topics of Chapter 6, namely, sufficiency,
information, and ancillarity, with nearly as much depth or breadth for the
target audience. In particular, Theorem 6.4.2 helps in proving the sufficiency
property of a statistic via its information content. The associated simple
examples and exercises then drive the point home. One will discover out-
of-the-ordinary remarks, ideas and examples throughout the book.
e) The history of statistics and statistical discoveries should not be sep-
arated from each other since neither can exist without the other. It may be
noted that Folks (1981) first added some notable historical remarks within
the material of his textbook written at the sophomore level. I have found
that at all levels of instructions, students enjoy the history very much and
they take more interest in the subject when the human element comes
alive. Thus, I have added historical remarks liberally throughout the text.
Additionally, in Section 14.2, I have given selected biographical notes on some
of the exceptional contributors to the development of statistics. The biogra-