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Interdisciplinary Ph.D. Program in Computational Analysis and Modeling (CAM)::Qualifying Exams::Probability and Statistics Axioms of Probability Random experiments, sample space and events, probability function, rules of probability Combinatorial Methods Permutations and combinations, ordered and unordered samples Random Variables Discrete random variables, continuous random variables, distribution functions, moments, probability generating function, special distributions: binomial, Poisson, hypergeometric geometric, negative Binomial, normal, Lognormal, negative exponential, uniform, Gamma and Chi-square, Beta Random Vectors Bivariate and multivariate distributions, multinomial distribution, marginal distributions and independence Distributions of Functions of Random Variables Sums of random variables, Jacobians, the t and F distributions, distributions of order statistics, expectations of functions of random variables Limit Theorems Chebyshev inequality and weak law of large numbers, strong law of large numbers, central limit theorem, convergence in distribution Conditional Distributions and Expectations Conditional densities and probability functions, conditional probability and independence, conditional expectations. Estimation Point estimation, bayesian estimates, confidence intervals for means and variances, sufficient statistics, maximum likelihood estimates, properties of maximum likelihood estimates, Rao-Cramer lower bound Statistical Hypotheses Certain best tests, uniformaly most powerful tests, likelihood ratio tests, sequential probability ratio test, Chi-square tests, T and F tests, noncentral F distributions, power of a test statistics, least squares, simple and multiple regression, analysis of variance Normal Distribution Theory The multivariate normal distribution, the distribution of centain quadratic forms, the independence of certain quadratic forms REFERENCES
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