Intermediate Probability: A Computational Approach

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

variables.

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.

References.

Index.