Study guide of topics covered for Midterm 2
ST 502
Rice textbook chapters covered: 1 - 9

--> Probability measures; definitions and properties
--> Conditional probability; definition and law of total probability
--> Bayes' rule, inverse probability 
--> Independence; definition and consequences
--> Random variables; discrete and continuous
--> Probability mass and density functions; defining properties
--> Cumulative distribution function; defining properties
--> Limit theorems
--> Notions of convergence; pointwise, in probability, in distribution, a.s.
--> Law of large numbers
--> Proof of the law of large numbers
--> Markov's inequality and proof
--> Central limit theorem
--> Taylor expansions
--> Moment generating functions and their relation to weak convergence
--> Standardizing a random variable
--> Monte Carlo sampling/integration; importance sampling
--> Distributions derived from the normal distribution; chi-squared, t, and F
--> Heavy tailed versus light tailed distributions
--> Change of variables and transformations of random variables
--> Properties of the sample mean and variance for iid Gaussian data
--> Population parameters versus sample statistics
--> Sampling distribution
--> Finite population sampling, with and without replacement
--> Finite population corrections to sample statistics (sampling without replacement)
--> Expected value and variance of sample statistics
--> Estimation of expected value and variance of sample statistics
--> Normal approximations 
--> Confidence intervals; derivation and interpretation

Topics after Midterm 1

--> Parameter estimation
--> Method of moments (MoM)
--> Population moments versus sample moments
--> Bootstrapping to estimate MoM sampling distributions
--> Definition of consistency of an estimator (in probability)
--> Method of maximum likelihood
--> Likelihood and log-likelihood function
--> Maximum likelihood estimate (MLE)
--> Large sample theory for MLE (consistency and asymptotic normality)
--> Fisher information
--> Exact and approximate confidence intervals for MLE
--> Concept of efficiency of an estimator
--> Mean squared error (MSE) and variance/bias tradeoff
--> Cramer-Rao lower bound
--> Definition of a sufficient statistic
--> Factorization theorem
--> The Bayesian approach
--> Prior and posterior distribution
--> Conjugate prior
--> Credible set/region
--> Conditional posterior distribution
--> Gibbs sampling
--> Metropolis-Hastings algorithm 
--> Hypothesis testing
--> Likelihood ratio (LR)
--> Prior and posterior odds ratio
--> Neyman-Pearson paradigm
--> Type I and type II error
--> Level of significance and power
--> LR test
--> Simple versus composite hypothesis
--> Neyman-Pearson lemma