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	Preface
	Contents
	Chapter 1	Introduction
	Chapter 2	Continuous Random Variables
	Chapter 3	Discrete Random Variables
	Chapter 4	Distributions of Functions of Random Variables
	Chapter 5	Systems of Distributions
	Chapter 6	Multivariate Distributions
	Chapter 7	Moments of Sampling Distributions
	Chapter 8	Asymptotic Theory
	Chapter 9	Statistical Decision Theory
	Chapter 10	Unbiased Parameter Estimation
	Chapter 11	Principles of Maximum Likelihood Estimation
	Chapter 12	Maximum Likelihood Estimation in Practice
	Appendix
	Notes
	References
	Index

Chapter 1: Introduction
1.1	Mathematical Statistics with Mathematica		1
	A	A New Approach	1
	B	Design Philosophy	1
	C	If You Are New to Mathematica	2
1.2	Installation, Registration and Password		3
	A	Installation, Registration and Password	3
	B	Loading mathStatica	5
	C	Help	5
1.3	Core Functions		6
	A	Getting Started	6
	B	Working with Parameters	8
	C	Discrete Random Variables	9
	D	Multivariate Random Variables	11
	E	Piecewise Distributions	13
1.4	Some Specialised Functions		15
1.5	Notation and Conventions		24
	A	Introduction	24
	B	Statistics Notation	25
	C	Mathematica Notation	27

Chapter 2: Continuous Random Variables
2.1	Introduction		31
2.2	Measures of Location		35
	A	Mean	35
	B	Mode	36
	C	Median and Quantiles	37
2.3	Measures of Dispersion		40
2.4	Moments and Generating Functions		45
	A	Moments	45
	B	The Moment Generating Function	46
	C	The Characteristic Function	50
	D	Properties of Characteristic Functions (and mgf's)	52
	E	Stable Distributions	56
	F	Cumulants and Probability Generating Functions	60
	G	Moment Conversion Formulae	62
2.5	Conditioning, Truncation and Censoring		65
	A	Conditional / Truncated Distributions	65
	B	Conditional Expectations	66
	C	Censored Distributions	68
	D	Option Pricing	70
2.6	Pseudo-Random Number Generation		72
	A	Mathematica's Statistics Package	72
	B	Inverse Method (Symbolic)	74
	C	Inverse Method (Numerical)	75
	D	Rejection Method	77
2.7	Exercises		80

Chapter 3: Discrete Random Variables
3.1	Introduction		81
3.2	Probability: 'Throwing' a Die		84
3.3	Common Discrete Distributions		89
	A	The Bernoulli Distribution	89
	B	The Binomial Distribution	91
	C	The Poisson Distribution	95
	D	The Geometric and Negative Binomial Distributions	98
	E	The Hypergeometric Distribution	100
3.4	Mixing Distributions		102
	A	Component-Mix Distributions	102
	B	Parameter-Mix Distributions	105
3.5	Pseudo-Random Number Generation		109
	A	Introducing DiscreteRNG	109
	B	Implementation Notes	113
3.6	Exercises		115

Chapter 4: Distributions of Functions of Random Variables

Chapter 5: Systems of Distributions
5.1	Introduction		149
5.2	The Pearson Family		149
	A	Introduction	149
	B	Fitting Pearson Densities	151
	C	Pearson Types	157
	D	Pearson Coefficients in Terms of Moments	159
	E	Higher Order Pearson-Style Families	161
5.3	Johnson Transformations		164
	A	Introduction	164
	B	SL System (Lognormal)	165
	C	SU System (Unbounded)	168
	D	SB System (Bounded)	173
5.4	Gram-Charlier Expansions		175
	A	Definitions and Fitting	175
	B	Hermite Polynomials; Gram-Charlier Coefficients	179
5.5	Non-Parametric Kernel Density Estimation		181
5.6	The Method of Moments		183
5.7	Exercises		185

Chapter 6: Multivariate Distributions
6.1	Introduction		187
	A	Joint Density Functions	187
	B	Non-Rectangular Domains	190
	C	Probability and Prob	191
	D	Marginal Distributions	195
	E	Conditional Distributions	197
6.2	Expectations, Moments, Generating Functions		200
	A	Expectations	200
	B	Product Moments, Covariance and Correlation	200
	C	Generating Functions	203
	D	Moment Conversion Formulae	206
6.3	Independence and Dependence		210
	A	Stochastic Independence	210
	B	Copulae	211
6.4	The Multivariate Normal Distribution		216
	A	The Bivariate Normal	216
	B	The Trivariate Normal	226
	C	CDF, Probability Calculations and Numerics	229
	D	Random Number Generation for the Multivariate Normal	232
6.5	The Multivariate t and Multivariate Cauchy		236
6.6	Multinomial and Bivariate Poisson		238
	A	The Multinomial Distribution	238
	B	The Bivariate Poisson	243
6.7	Exercises		248

Chapter 7: Moments of Sampling Distributions
7.1	Introduction		251
	A	Overview	251
	B	Power Sums and Symmetric Functions	252
7.2	Unbiased Estimators of Population Moments		253
	A	Unbiased Estimators of Raw Moments of the Population	253
	B	h-statistics: Unbiased Estimators of Central Moments	253
	C	k-statistics: Unbiased Estimators of Cumulants	256
	D	Multivariate h- and k-statistics	259
7.3	Moments of Moments		261
	A	Getting Started	261
	B	Product Moments	266
	C	Cumulants of k-statistics	267
7.4	Augmented Symmetrics and Power Sums		272
	A	Definitions and a Fundamental Expectation Result	272
	B	Application 1: Understanding Unbiased Estimation	275
	C	Application 2: Understanding Moments of Moments	275
7.5	Exercises		276

Chapter 8: Asymptotic Theory
8.1	Introduction		277
8.2	Convergence in Distribution		278
8.3	Asymptotic Distribution		282
8.4	Central Limit Theorem		286
8.5	Convergence in Probability		292
	A	Introduction	292
	B	Markov and Chebyshev Inequalities	295
	C	Weak Law of Large Numbers	296
8.6	Exercises		298

Chapter 9: Statistical Decision Theory
9.1	Introduction		301
9.2	Loss and Risk		301
9.3	Mean Square Error as Risk		306
9.4	Order Statistics		311
	A	Definition and OrderStat	311
	B	Applications	318
9.5	Exercises		322

Chapter 10: Unbiased Parameter Estimation
10.1	Introduction		325
	A	Overview	325
	B	SuperD	326
10.2	Fisher Information		326
	A	Fisher Information	326
	B	Alternate Form	329
	C	Automating Computation: FisherInformation	330
	D	Multiple Parameters	331
	E	Sample Information	332
10.3	Best Unbiased Estimators		333
	A	The Cramér-Rao Lower Bound	333
	B	Best Unbiased Estimators	335
10.4	Sufficient Statistics		337
	A	Introduction	337
	B	The Factorisation Criterion	339
10.5	Minimum Variance Unbiased Estimation		341
	A	Introduction	341
	B	The Rao-Blackwell Theorem	342
	C	Completeness and MVUE	343
	D	Conclusion	346
10.6	Exercises		347

Chapter 11: Principles of Maximum Likelihood Estimation

Chapter 12: Maximum Likelihood Estimation in Practice

Appendix
A.1	Is That the Right Answer, Dr Faustus?		421
A.2	Working with Packages		425
A.3	Working with = , ->, == and :=		426
A.4	Working with Lists		428
A.5	Working with Subscripts		429
A.6	Working with Matrices		433
A.7	Working with Vectors		438
A.8	Changes to Default Behaviour		443
A.9	Building Your Own mathStatica Function		446

Notes			447

References			463

Index			469