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Introduction to Probability TheoryChen, Yong-Yuan 25 May 2010 (has links)
In this paper, we first present the basic principles of set theory and combinatorial analysis which are the most useful tools in computing probabilities. Then, we show some important properties derived from axioms of probability. Conditional probabilities come into play not only when some partial information is available, but also as a tool to compute probabilities more easily, even when partial information is unavailable. Then, the concept of random variable and its some related properties are introduced. For univariate random variables, we introduce the basic properties of some common discrete and continuous distributions. The important properties of jointly distributed random variables are also considered. Some inequalities, the law of large numbers and the central limit theorem are discussed. Finally, we introduce additional topics the Poisson process.
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Statistical InferenceChou, Pei-Hsin 26 June 2008 (has links)
In this paper, we will investigate the important properties of three major parts of statistical inference: point estimation, interval estimation and hypothesis testing. For point estimation, we consider the two methods of finding estimators: moment estimators and maximum likelihood estimators, and three methods of evaluating estimators: mean squared error, best unbiased estimators and sufficiency and unbiasedness. For interval estimation, we consider the the general confidence interval, confidence interval in one sample, confidence interval in two samples, sample sizes and finite population correction factors. In hypothesis testing, we consider the theory of testing of hypotheses, testing in one sample, testing in two samples, and the three methods of finding tests: uniformly most powerful test, likelihood ratio test and goodness of fit test. Many examples are used to illustrate their applications.
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