## Probability EssentialsThis introduction to Probability Theory can be used, at the beginning graduate level, for a one-semester course on Probability Theory or for self-direction without benefit of a formal course; the measure theory needed is developed in the text. It will also be useful for students and teachers in related areas such as Finance Theory (Economics), Electrical Engineering, and Operations Research. The text covers the essentials in a directed and lean way with 28 short chapters. Assuming of readers only an undergraduate background in mathematics, it brings them from a starting knowledge of the subject to a knowledge of the basics of Martingale Theory. After learning Probability Theory from this text, the interested student will be ready to continue with the study of more advanced topics, such as Brownian Motion and Ito Calculus, or Statistical Inference. |

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### Contents

Introduction | 1 |

Probabilities on a Countable Space | 17 |

Construction of a Probability Measure | 31 |

Copyright | |

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assume Borel measurable Borel sets bounded Cauchy Central Limit Theorem characteristic function conditional expectation constant continuous functions converges in distribution converges weakly Corollary countable covariance matrix cr-algebra deduce defined Definition denote distribution function distribution measure distribution with parameters Dominated Convergence Theorem E{Xj E{Xn E{XY equivalent example Exercises for Chapter exponential fixed Fn(x function F fy(y Gamma Gaussian random variables given hence Hilbert space Hint hypothesis implies inequality integrable Large Numbers Law of Large Lebesgue measure Lemma let Q Let Xj)j>i Let Xn Let Xn)n>i lim^oo liminf linear martingale Monotone Convergence Theorem Moreover n-+oo nonnegative normal distribution Note open sets P(An P(Xn pairwise disjoint Poisson positive probability measure probability space prove px(u result Rn-valued sequence of random Show simple Strong Law sub a-algebra submartingale subset Suppose unique valued random variables vector weak convergence Xn converges