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Intermediate Probability: A Computational Approach

Marc S. Paolella
John Wiley
Publication Date: 
Number of Pages: 
[Reviewed by
Miklós Bóna
, on

The book is a sequel to Fundamental Probability , by the same author. It has three main topics: sums and other functions of random variables, asymptotics, and univariate distributions. 

The first part covers generating functions, Laplace and Fourier Transforms, inversion formulas, and ends with multivariate normal distributions. The treatment is self-contained, that is, all the notions that an advanced undergraduate would not necessarily know are defined and explained.

In the section on asymptotics and other approximations the author goes through the various notions of convergence, and establishes their order of strength. This reviewer would have liked to see techniques to prove convergence in applications, such as proving asymptotic normality or convergence to a Poisson distribution in combinatorics, by classic techniques such as the method of moments. Other subjects covered in this part are saddlepoint approximations, and order statistics, in both the univariate and the multivariate case.

The third, and most advanced part, covers a selection of univariate distributions and their generalizations. They include weighted sums of independent random variables, the stable Paretian distribution, hyperbolic, and Gaussian distributions and their generalizations, and various noncentral distributions.

As the subtitle suggests, great emphasis is given to numerically computing several examples, and to the various methods needed to achieve that. This is particularly so in the last part of the book, where the Paretian distribution, and various noncentral distributions are treated. Numerous codes in Maple and R are included.

The reader-friendly style of the text itself would make the book appropriate for self-study or classroom adoption, but it is strange that no exercises come with as much as a numerical answer, let alone with full solutions. This reviewer did not find any such information on the book's website either. Students need some way to check whether their answers are correct.

Miklós Bóna is Associate Professor of Mathematics at the University of Florida.


I Sums of Random Variables.

1 Generating functions.

1.1 The moment generating function.

1.2 Characteristic functions.

1.3 Use of the fast Fourier transform.

1.4 Multivariate case.

1.5 Problems.

2 Sums and other functions of several random variables.

2.1 Weighted sums of independent random variables.

2.2 Exact integral expressions for functions of two continuous random


2.3 Approximating the mean and variance.

2.4 Problems.

3 The multivariate normal distribution.

3.1 Vector expectation and variance.

3.2 Basic properties of the multivariate normal.

3.3 Density and moment generating function.

3.4 Simulation and c.d.f. calculation.

3.5 Marginal and conditional normal distributions.

3.6 Partial correlation.

3.7 Joint distribution of Xbar and S2 for i.i.d. normal samples.

3.8 Matrix algebra.

3.9 Problems.

II Asymptotics and Other Approximations.

4 Convergence concepts.

4.1 Inequalities for random variables.

4.2 Convergence of sequences of sets.

4.3 Convergence of sequences of random variables.

4.4 The central limit theorem.

4.5 Problems.

5 Saddlepoint approximations.

5.1 Univariate.

5.2 Multivariate.

5.3 The hypergeometric functions 1F1 and 2F1.

5.4 Problems.

6 Order statistics.

6.1 Distribution theory for i.i.d. samples.

6.2 Further examples.

6.3 Distribution theory for dependent samples.

6.4 Problems.

III More Flexible and Advanced Random Variables.

7 Generalizing and mixing.

7.1 Basic methods of extension.

7.2 Weighted sums of independent random variables.

7.3 Mixtures.

7.4 Problems.

8 The stable Paretian distribution.

8.1 Symmetric stable.

8.2 Asymmetric stable.

8.3 Moments.

8.4 Simulation.

8.5 Generalized central limit theorem.

9 Generalized inverse Gaussian and generalized hyperbolic distributions.

9.1 Introduction.

9.2 The modified Bessel function of the third kind.

9.3 Mixtures of normal distributions.

9.4 The generalized inverse Gaussian distribution.

9.5 The generalized hyperbolic distribution.

9.6 Properties of the GHyp distribution family.

9.7 Problems.

10 Noncentral distributions.

10.1 Noncentral chi-square.

10.2 Singly and doubly noncentral F.

10.3 Noncentral beta.

10.4 Singly and doubly noncentral t.

10.5 Saddlepoint uniqueness for the doubly noncentral F.

10.6 Problems.

A Notation and distribution tables.