Outline

1.

Probability and Univariate Distribution Theory

(a)

Probability axioms, counting methods, equally likely outcomes; law of total probability and Bayes theorem, conditional probability and independence.

(b)

Properties of univariate random variables, density and mass functions, distribution functions. Exponential family of distributions as well as these distributions: uniform discrete, binomial, hypergeometric, Poisson, negative binomial. normal, exponential, gamma, chi-square, T, F, and beta.

(c)

Moments and expected value, moment generating functions.

(d)

Approximate methods using Taylor series approximations (delta method).

(e)

Functions of random variables: CDF method, change-of-variable method.

2.

Multivariate distributions:

(a)

Joint and marginal distributions, conditional distributions, independent random variables.

(b)

Transformations of random vectors, expected values, covariance and correlation.

(c)

Properties of linear combinations of random variables .

(d)

Properties of multinomial and multivariate normal distributions.

(e)

Hierarchical models and mixtures.

3.

Random samples and statistics

(a)

Sampling from normal distribution: distribution of sample mean and variance; Chi-square, T and F distributions

(b)

Distribution of order statistics

(c)

Sufficient statistics, ancillary statistics

4.

Large Sample Methods

(a)

Sequence of random variables

(b)

Convergence in distribution, convergence in probability

(c)

Limiting moment-generating functions

(d)

Central limit theorem

(e)

Approximate methods using Taylor series approximations (delta method).

5.

Parameter Estimation

(a)

Location, scale, and shape parameters

(b)

Method of moments, maximum likelihood , Bayes estimators

(c)

Evaluating estimators: bias, variance, mean-squared-error, invariance, consistent estimators.

(d)

Asymptotic properties of maximum-likelihood estimators.

(e)

Taylor series methods (delta method) for approximating properties of estimators.

6.

Hypothesis testing

(a)

Evaluating tests: power, unbiased tests, uniformly most powerful tests

(b)

Methods of finding tests: Neyman-Pearson lemma, likelihood ratio tests

(c)

Some standard tests: tests on means and variances from a normal distribution, two-sample t-test, F-test comparing variances, paired difference t-test; tests on one and two binomial proportions, chi-square tests for goodness-of-fit, homogeneity, and independence.

(d)

Non-central distributions and power computations.

7.

Interval Estimation

(a)

Inverting a hypothesis test

(b)

Pivotal quantities

(c)

Evaluating intervals: size and coverage rates.

(d)

Approximate maximum likelihood estimate based intervals