AN INTRODUCTION TO APPLIED PROBABILITY
CONTENTS BASIC NOTIONS OUTCOMES AND EVENTS PROBABILITY OF EVENTS INDEPENDENCE AND CONDITIONING COUNTING MODELS EXERCISES DISCRETE RANDOM VARIABLES PROBABILITY DISTRIBUTION AND EXPECTATION REMARKABLE DISCRETE DISTRIBUTIONS GENERATING FUNCTIONS CONDITIONAL EXPECTATION I CONTINUOUS RANDOM VECTORS RANDOM VARIABLES WITH REAL VALUES SQUARE-INTEGRABLE RANDOM VARIABLES GAUSSIAN VECTORS CONDITIONAL EXPECTATION II THE LEBESGUE INTEGRAL MEASURABLE FUNCTIONS AND MEASURES THE INTEGRAL BASIC PROPERTIES OF THE INTEGRAL THE BIG THEOREMS SOLUTIONS FROM INTEGRAL TO EXPECTATION TRANSLATION THE DISTRIBUTION OF A RANDOM ELEMENT CHARACTERISTIC FUNCTIONS INDEPENDENCE CONDITIONAL EXPECTATION III GENERAL THEORY OF CONDITIONAL EXPECTATION CONVERGENCE ALMOST SURE A SU?CIENT CONDITION AND A CRITERION THE STRONG LAW OF LARGE NUMBERS KOLMOGOROV’S ZERO-ONE LAW RELATED TYPES OF CONVERGENCE UNIFORM INTEGRABILITY CONVERGENCE IN DISTRIBUTION PAUL L´EVY’S CRITERION THE CENTRAL LIMIT THEOREM CONVERGENCE IN VARIATION THE RANK OF CONVERGENCE IN DISTRIBUTION MARTINGALES THE MARTINGALE PROPERTY MARTINGALE INEQUALITIES THE OPTIONAL SAMPLING THEOREM THE MARTINGALE CONVERGENCE THEOREM SQUARE-INTEGRABLE MARTINGALES MARKOV CHAINS THE TRANSITION MATRIX RECURRENCE LONG-RUN BEHAVIOR ABSORPTION THE MARKOV PROPERTY ON GRAPHS MONTE CARLO MARKOV CHAINS 10 POISSON PROCESSES 10 1 POISSON PROCESSES ON THE LINE 10 2 GENERALITIES ON POINT PROCESSES 10 3 SPATIAL POISSON PROCESSES 10 4 OPERATIONS ON POISSON PROCESSES 10 5 EXERCISES 11 BROWNIAN MOTION 11 1 CONTINUOUS-TIME STOCHASTIC PROCESSES 11 2 GAUSSIAN PROCESSES 11 3 THE WIENER–DOOB INTEGRAL 11 4 TWO APPLICATIONS 11 5 FRACTAL BROWNIAN MOTION 11 6 EXERCISES 12 WIDE-SENSE STATIONARY PROCESSES 12 1 THE POWER SPECTRAL MEASURE 12 2 FILTERING OF WSS STOCHASTIC PROCESSES 12 3 THE CRAM´ER–KHINCHIN DECOMPOSITION 12 4 MULTIVARIATE WSS STOCHASTIC PROCESSES 12 5 EXERCISES