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Course notes for Mathematical Statistics (APPM/STAT/MATH 4520/5520), Fall 2017, University of Colorado, Boulder taught by Professor Jem N. Corcoran.
Cover
Preface
Probability Preliminaries
Between Zero and One
Counting
Sample Spaces, Events, and Some Simple Probabilities
Some Very Brief Words About Independence and "Disjointness"
A Brief Review of Random Variables, PDFs, and CDFs
The Expected Value, Expectation, or Mean of a Random Variable or a Distribution
The Variance of a Random Variable or a Distribution
Indicator Notation and a Most Useful Appendix
Joint PDFs, Marginals, and Independence
Useful Properties of Expectation, Variance, and Covariance
MathStat Preliminaries: Four Important Tools for Mathematical Statistics
Wait. Where are we going?
"Integrating Without Integrating", A.K.A. The Best Trick You'll Learn Today
Important Tool I: Finding Distributions of Transformations of Random Variables
Important Tool II: Bivariate Transformations
Important Tool II: Mins and Maxes
Important Tool IV: Moment Generating Functions (MGFs)
Qualities of Estimators: Defining Good, Better, and Best
Notation, Statistics, and Unbiasedness
Mean Squared Error and Bias
Convergence in Probability
Convergence In Distribution
The Central Limit Theorem
A "Mostly Normal" Introduction to Confidence Intervals
A Confidence Interval for the Mean of a Normal Distribution: σ2 is Known
An Approximate Confidence Interval for the Mean of Any Distribution
A Confidence Interval for the Mean of a Normal Distribution: σ2 is Unknown and Sample Size is Small
A Difference of Means
A "Mostly Normal" Introduction to Hypothesis Testing
Errors in Hypothesis Testing
Finding a Test: A Simple Null Hypothesis
Finding a Test: A Composite Null Hypothesis
Power
Estimation
Method of Moments Estimators (MMEs)
Maximum Likelihood Estimators (MLEs)
The Cramer-Rao Lower Bound (CRLB)
Asymptotic Properties of MLEs
Uniformly Minimum Variance Unbiased Estimators (UMVUEs)
General Hypothesis Testing
A Best Test
Uniformly Most Powerful Tests (UMPs)
Generalized Likelihood Ratio Tests (GLRTs)
Tables of Distributions
The Jacobian and a Change of Variables
The Jacobian
A Bivariate Transformation
More Variables
Standard Normal, t, X2 and F Tables
Symbols/Notation
Index