UT Tyler Department of Mathematics

Master's Exam Syllabi

Probability and Statistics Sequence

There is more listed here than can reasonably be covered in a two semester sequence. Topics for the masters oral exam will be drawn from those actually covered in class.

Probability

1. Basic Probability
   1. Probability Axioms
   2. Probability Spaces
   3. Set Theoretic Probability
   4. Conditional Probability
   5. Bayes' Theorem
2. Discrete Random Variables
   1. Some basic discrete r.v.s
      1. Discrete Uniform
      2. Bernoulli
      3. Binomial
      4. Geometric
      5. Negative Binomial
      6. Hypergeometric
      7. Poisson
   2. Moments, moment generating functions.
   3. Expectations, variances
   4. Distribution of sums, quotients, etc.
3. Continuous Random Variables
   1. Some basic continuous r.v.s
      1. Continuous Uniform
      2. Normal
      3. Chi-squared
      4. F
      5. Student's T
      6. Gamma
      7. Exponential
   2. Expectations, variances
   3. Distribution of sums, quotients, etc.
4. Connections to Statistics
   1. Sampling distributions when sampling from normal populations
   2. Bayesian prior/posterior distributions
5. Poisson Processes
6. Limits of random variables, convergence in probability.
7. Central Limit Theorem
8. Other topics covered in class as appropriate.

Statistics

1. Point Estimates for population parameters
   1. Maximum Likelihood
   2. Method of Moments
   3. Bias
   4. Exponential families
   5. Fisher factorization
   6. Crámer Rao lower bound
   7. UMVUE
2. Confidence Intervals for population parameters
3. Hypothesis testing
   1. Neyman-Pearson Lemma
   2. Likelihood ratio tests
4. Linear Regression
   1. Linear Models and assumptions made
   2. Least Squares Estimation
   3. Confidence Intervals and Hypothesis Tests for model parameters
   4. Applications, residual analysis, etc.
   5. Generalizations (weighted least squares, etc.)
5. ANOVA
   1. Relationship to linear models
   2. Multiple comparisons
6. Other topics covered in class as appropriate.