I. INTRODUCTION

Definitions of sample spaces, subsets, and events. Elementary operations on sets and the algebra of sets. Set functions.

2. PROBABILITY

The mathematics of probability-rules and theorems. Conditional probability, independent events and Bayes rule, inclusion-exclusion.

3. PROBABILITY FUNCTIONS

Definitions of a random variable and its probability function. Discrete probability functions including: binomial, hypergeometric, geometric, negative binomial and Poisson. Multivariate discrete distributions.

4. PROBABILITY DENSITIES

Definition of a density function. Continuous density functions including: uniform, exponential, gamma, beta and normal. Change of variables. Multivariate densities including discussion of joint, marginal and conditional densities.

5. MATHEMATICAL EXPECTATION

Definition of moments. Chebychev's Inequality. Moments of distributions previously discussed. Moment generating functions, Markov inequality. Discussion of product moments; covariance and relation to independence, moments of Linear Combinations of Random Variables. Conditional Expectations.

6. FUNCTIONS OF RANDOM VARIABLES

Distribution of functions of random variables via convolutions, moment generating functions and transformation of variables. Distribution of the mean for a finite population. Central limit theorem. Probability integral transformation.

7. SAMPLING DISTRIBUTION

Distribution of statistics; sample mean and sample variance. The chi-squared, F and t distributions. Sampling distribution of order statistics. Joint distribution of order statistics.

8. ESTIMATION AND CONFIDENCE INTERVALS

Some properties of point estimators, some common unbiased point estimators, evaluating the goodness of a point estimator, confidence intervals, large sample confidence interval, selecting the sample size, small-sample confidence interval.

9. PROPERTIES OF POINT ESTIMATORS AND METHOD OF ESTIMATION

Unbiasedness, relative efficiency, consistency, minomial sufficiency and minimum variance unbiased estimation (MVUE), the method of moments, the method of maximum likelihood.

10. HYPOTHESIS TESTING

Elements of a statistical test, common large-sample tests, calculation of type-I error, sample size for the Z-test, different ways of reporting the result of a test, attained significance levels or p-values, some comments on the theory of hypothesis testing, two-sample tests based on t-distributions, testing hypothesis concerning variances, power of tests, the Neyman-Pearson lemma, likelihood ratio tests. Analysis of contingency table, the Chi-Square goodness of fit test.

11. REGRESSION AND CORRELATION GENERAL LINEAR MODEL FORMULATION (optional)

Linear regression and estimation by least squares method, method of least squares and its connection with maximum likelihood method under normality assumption, confidence intervals for parameters, prediction interval, Multiple Linear regression.

12. ANALYSIS OF VARIANCE (optional)

Experimental Design, one-way and two-way analysis of variance.

13. NONPARAMETRIC TESTS (optional)

Sign test, Signed-Rank test, Rank-Sum tests, Tests based on Runs, Rank correlation.