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. |
Contents
Introduction | 1 |
Probabilities on a Countable Space | 17 |
Exercises | 27 |
Copyright | |
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assume Binomial Borel sets bounded Cauchy Central Limit Theorem characteristic function conditional expectation constant continuous functions converges in distribution converges weakly Corollary countable covariance matrix deduce defined Definition denote density distribution function Dominated Convergence Theorem E{X² E{Xn E{Xo E{XY example Exercises for Chapter exponential finite Fn(x function F Gamma Gaussian random variables given hence Hilbert space Hint implies independent inequality Large Numbers Law of Large Lebesgue measure Lemma Let ƒ let G Let Sn Let X1 lim inf lim sup limn linear martingale Monotone Convergence Theorem nonnegative Note P(An P(Xn Poisson probability measure probability space R-valued result sequence of random Show Sn(x Strong Law sub o-algebra submartingale subset subspace supermartingale Suppose types of convergence unique valued random variables variance vector weak convergence Xn converges Y₁ Y₂ σ²