PROBABILITY AND STATISTICS BY EXAMPLE: BASIC PROBABILITY AND STATISTICS

By WILLIAM WILLIAMS

CONTENTS PART I BASIC PROBABILITY DISCRETE OUTCOMES A UNIFORM DISTRIBUTION CONDITIONAL PROBABILITIES THE BAYES THEOREM INDEPENDENT TRIALS THE EXCLUSION–INCLUSION FORMULA THE BALLOT PROBLEM RANDOM VARIABLES EXPECTATION AND CONDITIONAL EXPECTATION JOINT DISTRIBUTIONS THE BINOMIAL, POISSON AND GEOMETRIC DISTRIBUTIONS PROBABILITY GENERATING, MOMENT GENERATING AND CHARACTERISTIC FUNCTIONS CHEBYSHEV’S AND MARKOV’S INEQUALITIES JENSEN’S INEQUALITY THE LAW OF LARGE NUMBERS AND THE DE MOIVRE–LAPLACE THEOREM BRANCHING PROCESSES CONTINUOUS OUTCOMES UNIFORM DISTRIBUTION PROBABILITY DENSITY FUNCTIONS RANDOM VARIABLES INDEPENDENCE EXPECTATION, CONDITIONAL EXPECTATION, VARIANCE, GENERATING FUNCTION, CHARACTERISTIC FUNCTION NORMAL DISTRIBUTIONS CONVERGENCE OF RANDOM VARIABLES AND DISTRIBUTIONS THE CENTRAL LIMIT THEOREM PART II BASIC STATISTICS PARAMETER ESTIMATION PRELIMINARIES SOME IMPORTANT PROBABILITY DISTRIBUTIONS ESTIMATORS UNBIASEDNESS SUFFICIENT STATISTICS THE FACTORISATION CRITERION MAXIMUM LIKELIHOOD ESTIMATORS NORMAL SAMPLES THE FISHER THEOREM MEAN SQUARE ERRORS THE RAO–BLACKWELL THEOREM THE CRAMÉR–RAO INEQUALITY EXPONENTIAL FAMILIES CONFIDENCE INTERVALS BAYESIAN ESTIMATION HYPOTHESIS TESTING TYPE I AND TYPE II ERROR PROBABILITIES MOST POWERFUL TESTS LIKELIHOOD RATIO TESTS THE NEYMAN–PEARSON LEMMA AND BEYOND GOODNESS OF FIT TESTING NORMAL DISTRIBUTIONS, 1: HOMOGENEOUS SAMPLES THE PEARSON 2 TEST THE PEARSON THEOREM GENERALISED LIKELIHOOD RATIO TESTS THE WILKS THEOREM CONTINGENCY TABLES TESTING NORMAL DISTRIBUTIONS, 2: NON-HOMOGENEOUS SAMPLES LINEAR REGRESSION THE LEAST SQUARES ESTIMATORS LINEAR REGRESSION FOR NORMAL DISTRIBUTIONS CAMBRIDGE UNIVERSITY MATHEMATICAL TRIPOS EXAMINATION QUESTIONS IN IB STATISTICS (1992–1999)