Intermediate Mathematical Statistics I

Description:

Sample spaces, probability and conditional probability, independence, random variables, expectation, distribution theory, sampling distributions, laws of large numbers and asymptotic theory, order statistics.

Required Book:

Statistical Inference (Second edition) by G. Casella and R.L. Berger. Duxbury, 2002.

Material: This material is mostly from Chapters 1-5 of the textbook

1. Probability Theory:

Set Theory. Probability Theory. Conditional Probability and Independence. Random Variables. Distribution Functions. Density and Mass Functions.

2. Transformations and Expectations:

Distribution of Functions of a Random Variable. Expected Values. Moments and Moment Generating Functions. Differentiating Under an Integral Sign.

3. Common Families of Distributions:

Introductions. Discrete Distributions. Continuous Distributions. Exponential Families. Locations and Scale Families. Inequalities and Identities.

4. Multiple Random Variables: Joint and Marginal Distributions. Conditional Distributions and Independence. Bivariate Transformations. Hierarchical Models and Mixture

Distributions. Covariance and Correlation. Multivariate Distributions. Inequalities.

5. Properties of a Random Sample:

Basic Concepts of Random Samples. Sums of Random Variables from a Random Sample. Sampling for the Normal Distribution. Order Statistics. Convergence Concepts. Generating a Random Sample.