CAM Ph.D. Qualifying Exam Contents


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

  • R.V. Hogg and A.T. Craig, Introduction to Mathematical Statistical, Fourth Edition, Macnillian.
  • A.M. Mood and F.A. Graybill, Introduction to the Theory of Statistics, McGraw Hill.
  • S. Ghahramani, Fundamentals of Probability, Prentice Hall.
  • M. Woodroofe, Probability with Applications, McGraw Hill.


    • CAM Ph.D. Program