Notes

Likelihood Ratio Tests and Chi-Squared Tests

Hypothesis Testing Beyond Nomality

t-Tests and Two-Sample Tests

Composite Test - Power Functions - and P-Values

Fundamental Concepts of Hypothesis Testing

Confidence Intervals Beyond the Normal Distribution

Confidence Intervals Involving the Normal Distribution

Likelihood Estimation

Point Estimation

Central Limit Theorem

Joint Distributions and Covariance

Continuous Random Variables

Discrete Random Variable

Conditional Probability

Descriptive Statistics and the Axioms of Probability

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Likelihood Ratio Tests and Chi-Squared Tests

• Independence and Homogeneity
• Wilks' Theorem Exploration
• Chi-Squared Goodness of Fit Test
• Wilks' Theorem for Large Sample GLRTs
• The Generalized Likelihood Ratio Test
• A Review of Maximum Likelihood Estimation

Hypothesis Testing Beyond Nomality

• The F-Distribution and a Ratio of Variances
• A Test for the Variance of the Normal Distribution
• Uniformly Most Powerful Tests
• The Neyman-Pearson Lemma - The Best Test
• Two Hypothesis Tests for the Rate of an Exponential Distribution
• Properties of the Exponential Distribution

t-Tests and Two-Sample Tests

• Comparing Population Proportions
• Welch's t-Test and Paired Data
• t-Tests and Two Sample Tests
• Two-Sample t-Tests for a Difference in Two Population Means
• Two-sample Tests Involving Means of Normal Distributions
• The t Test
• The Sample Variance for the Normal Distribution
• The t and Chi-Squared Distribution

Composite Test - Power Functions - and P-Values

• Hypothesis Tests for Proportions
• Two-Tailed Tests for the Mean of a Normal Distribution
• Estimating the Distributions of P-Values
• Power Functions
• P-values and QQ Plots
• One-Tailed Tests for the Mean of a Normal Distribution
• Composite Hypotheses and Level of Significance

Fundamental Concepts of Hypothesis Testing

• Test Statistics and Significance
• A First Test
• Visualizing Errors in Hypothesis Testing
• Normal Computations
• Errors in Hypothesis Testing
• Types of Hypotheses

Confidence Intervals Beyond the Normal Distribution

• Non-normal Confedence Intervals
• General Confidence Intervals Part 2
• General Confidence Intervals Part 1
• A Confidence Interval for a Ratio of Variances
• Confidence Intervals for Variances and a Difference of Proportions
• A Confidence Interval for Proportions

Confidence Intervals Involving the Normal Distribution

• t Distribution Confidence Interval
• Small Sample Confidence Intervals for the Difference Between Population Means
• Confidence Interval in R
• Confidence Intervals for The Difference Between Population Means
• Exploring The Normal, t and Chi-Squared Relationships
• The t and Chi-Squared Distributions and The Sample Variance
• A Normal Introduction to Confidence Intervals

Likelihood Estimation

• Large Sample Properties of MLEs
• The Weak Law of Large Numbers
• The Central Limit Theorem
• Central Limit Theorem Exercise
• Computational Simplifications for the CRLB
• Fisher Information and the Cramér-Rao Lower Bound
• The invariance property
• Mean square error, bias, and relative efficiency
• Multiple parameters and parameters in the support of a distribution a
• Notation and Terminology

Point Estimation

• Variance and Covariance
• Transformations of Distributions
• Table Summarizing Important Distributions
• Method of Moments Estimators
• Important Continous Distributions
• Expectation and Properties of Expectation
• Estimators and Sampling Distributions
• Distributions of Sums
• Important Discrete Distributions

Central Limit Theorem

• Central Limit Theorem Examples
• Introduction to the Central Limit Theorem

Joint Distributions and Covariance

• Covariance and Correlation
• Jointly Distributed Random Variables
• More on Expectation and Variance

Continuous Random Variables

• The Gaussian (normal) Random Variable Part 2
• The Gaussian (normal) Random Variable Part 1
• The Poisson and Exponential Random Variables
• Continuous Random Variables

Discrete Random Variable

• Binomial and Negative Binomial Random Variables
• Expectation and Variance
• Bernoulli and Geometric Random Variables
• Discrete Random Variable

Conditional Probability

• Independent Events
• Conditional Probability and Bayes Theorem

Descriptive Statistics and the Axioms of Probability

• Counting - Permutations and Combinations
• Axioms of Probability
• Intro to Probability