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Fundamental Probability: A Computational Approach

Marc S. Paolella
John Wiley
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
, on

Fundamental Probability: A Computational Approach is designed for students who have studied calculus and linear algebra and assumes no previous background in probability. However, because of the amount of material and the depth of the treatment, it is most appropriate for beginning graduate students or strong upper-division undergraduates. The intended audience includes students of mathematics, statistics, biology, computer science and engineering, but the book is tuned somewhat for those with interests in the quantitative aspects of finance and econometrics. The author’s goals are to focus on the practical aspects of the subject by emphasizing examples and computation, to push beyond the ordinary topics, and to include valuable techniques that are often omitted from introductory texts because they are computationally intensive.

The progression of topics is more or less conventional, although the approach and level of depth in each area are often unconventional. For example, the treatment of combinatorics is a good deal more extensive than comparable texts, and combinatorial arguments are used broadly throughout. The author treats the four basic discrete sampling schemes — with and without replacement, with and without a fixed number of draws determined in advance — in careful detail, and derives, for example, the moments of the distribution for sampling without replacement until r successes occur. This is done first in the univariate setting and is repeated in a later chapter for the multivariate case. Occupancy problems are also treated fairly extensively, and follow the author’s pattern of a mixture of combinatorial, computational, and analytic techniques.

The author takes pains to help the student develop tools for evaluating and understanding new continuous probability distributions: identifying parameters, recognizing major applications, investigating behavior of the tails. By examples the reader is introduced to basic concepts used in finance including utility functions, stochastic dominance, value at risk, and expected shortfall. MATLAB programs used throughout the text help to integrate the computational and analytical. These MATLAB routines could easily be converted to Maple or a comparable language without much difficulty. Exercises are well-chosen, plentiful and graded by degree of difficulty. The publisher’s website for the text has a complete set of solutions available online.

One of the more unusual aspects of this text is the lengthy appendix (more than ninety pages) on “calculus review”. The author wishes to “separate mathematics from the probabilistic concepts and distributional results obtained in the text”. He evidently expects students to review the appendix before beginning work on Chapter 1. I wonder how well this works in practice.

Overall, the text is polished and thorough with a clear development and many good examples.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.


A note to the student (and instructor).

A note to the instructor (and student).




1. Combinatorics.

1.1 Basic counting.

1.2 Generalized binomial coefficients.

1.3 Combinatoric identities and the use of induction.

1.4 The binomial and multinomial theorems.

1.4.1 The binomial theorem.

1.4.2 An extension of the binomial theorem.

1.4.3 The multinomial theorem.

1.5 The gamma and beta functions.

1.5.1 The gamma function.

1.5.2 The beta function.

1.6 Problems.

2. Probability spaces and counting.

2.1 Introducing counting and occupancy problems.

2.2 Probability spaces.

2.2.1 Introduction.

2.2.2 Definitions.

2.3 Properties.

2.3.1 Basic properties.

2.3.2 Advanced properties.

2.3.3 A theoretical property.

2.4 Problems.

3. Symmetric spaces and conditioning.

3.1 Applications with symmetric probability spaces.

3.2 Conditional probability and independence.

3.2.1 Total probability and Bayes’ rule.

3.2.2 Extending the law of total probability.

3.2.3 Statistical paradoxes and fallacies.

3.3 The problem of the points.

3.3.1 Three solutions.

3.3.2 Further gambling problems.

3.3.3 Some historical references.

3.4 Problems.


4. Univariate random variables.

4.1 Definitions and properties.

4.1.1 Basic definitions and properties.

4.1.2 Further definitions and properties.

4.2 Discrete sampling schemes.

4.2.1 Bernoulli and binomial.

4.2.2 Hypergeometric.

4.2.3 Geometric and negative binomial.

4.2.4 Inverse hypergeometric.

4.2.5 Poisson approximations.

4.2.6 Occupancy distributions.

4.3 Transformations.

4.4 Moments.

4.4.1 Expected value of X.

4.4.2 Higher-order moments.

4.4.3 Jensen?s inequality.

4.5 Poisson processes.

4.6 Problems.

5. Multivariate random variables.

5.1 Multivariate density and distribution.

5.1.1 Joint cumulative distribution functions.

5.1.2 Joint probability mass and density functions.

5.2 Fundamental properties of multivariate random variables.

5.2.1 Marginal distributions.

5.2.2 Independence.

5.2.3 Exchangeability.

5.2.4 Transformations.

5.2.5 Moments.

5.3 Discrete sampling schemes.

5.3.1 Multinomial.

5.3.2 Multivariate hypergeometric.

5.3.3 Multivariate negative binomial.

5.3.4 Multivariate inverse hypergeometric.

5.4 Problems.

6. Sums of random variables.

6.1 Mean and variance.

6.2 Use of exchangeable Bernoulli random variables.

6.2.1 Examples with birthdays.

6.3 Runs distributions.

6.4 Random variable decomposition.

6.4.1 Binomial, negative binomial and Poisson.

6.4.2 Hypergeometric.

6.4.3 Inverse hypergeometric.

6.5 General linear combination of two random variables.

6.6 Problems.


7. Continuous univariate random variables.

7.1 Most prominent distributions.

7.2 Other popular distributions.

7.3 Univariate transformations.

7.3.1 Examples of one-to-one transformations.

7.3.2 Many-to-one transformations.

7.4 The probability integral transform.

7.4.1 Simulation.

7.4.2 Kernel density estimation.

7.5 Problems.

8. Joint and conditional random variables.

8.1 Review of basic concepts.

8.2 Conditional distributions.

8.2.1 Discrete case.

8.2.2 Continuous case.

8.2.3 Conditional moments.

8.2.4 Expected shortfall.

8.2.5 Independence.

8.2.6 Computing probabilities via conditioning.

8.3 Problems.

9. Multivariate transformations.

9.1 Basic transformation.

9.2 The t and F distributions.

9.3 Further aspects and important transformations.

9.4 Problems.

Appendix A. Calculus review.

A.0 Recommended reading.

A.1 Sets, functions and fundamental inequalities.

A.2 Univariate calculus.

A.2.1 Limits and continuity.

A.2.2 Differentiation.

A.2.3 Integration.

A.2.4 Series.

A.3 Multivariate calculus.

A.3.1 Neighborhoods and open sets.

A.3.2 Sequences, limits and continuity.

A.3.3 Differentiation.

A.3.4 Integration.

Appendix B. Notation tables.

Appendix C. Distribution tables.