Page 10 - Applied Statistics Using SPSS, STATISTICA, MATLAB and R

Contents xi

Appendix A - Short Survey on Probability Theory 403

A.1 Basic Notions..........................................................................................403
A.1.1 Events and Frequencies .............................................................403
A.1.2 Probability Axioms....................................................................404
A.2 Conditional Probability and Independence .............................................406
A.2.1 Conditional Probability and Intersection Rule...........................406
A.2.2 Independent Events ...................................................................406
A.3 Compound Experiments..........................................................................408
A.4 Bayes’ Theorem......................................................................................409
A.5 Random Variables and Distributions ......................................................410
A.5.1 Definition of Random Variable .................................................410
A.5.2 Distribution and Density Functions...........................................411
A.5.3 Transformation of a Random Variable......................................413
A.6 Expectation, Variance and Moments ......................................................414
A.6.1 Definitions and Properties .........................................................414
A.6.2 Moment-Generating Function ...................................................417
A.6.3 Chebyshev Theorem..................................................................418
A.7 The Binomial and Normal Distributions.................................................418
A.7.1 The Binomial Distribution.........................................................418
A.7.2 The Laws of Large Numbers.....................................................419
A.7.3 The Normal Distribution ...........................................................420
A.8 Multivariate Distributions.......................................................................422
A.8.1 Definitions.................................................................................422
A.8.2 Moments....................................................................................425
A.8.3 Conditional Densities and Independence...................................425
A.8.4 Sums of Random Variables.......................................................427
A.8.5 Central Limit Theorem..............................................................428

Appendix B - Distributions 431

B.1 Discrete Distributions .............................................................................431
B.1.1 Bernoulli Distribution................................................................431
B.1.2 Uniform Distribution.................................................................432
B.1.3 Geometric Distribution..............................................................433
B.1.4 Hypergeometric Distribution.....................................................434
B.1.5 Binomial Distribution................................................................435
B.1.6 Multinomial Distribution...........................................................436
B.1.7 Poisson Distribution ..................................................................438
B.2 Continuous Distributions ........................................................................439
B.2.1 Uniform Distribution.................................................................439
B.2.2 Normal Distribution...................................................................441
B.2.3 Exponential Distribution............................................................442
B.2.4 Weibull Distribution..................................................................444
B.2.5 Gamma Distribution ..................................................................445
B.2.6 Beta Distribution .......................................................................446
B.2.7 Chi-Square Distribution.............................................................448