Fundamentals of Probability

1. Fundamentals
   1. Review of elementary probability: conditional probability, independence, random variables, expectations, conditional expectation
   2. Probability triples: Lebesgue-Stieltjes measure and product measure
   3. Random variables: cdf and pdf, independence, expectation, calculational tools, stochastic processes
2. Generating Functions and Inequalities
   1. Moments and moment generating function
   2. Inequalities: Markov, Chebyshev, Jensen, Holder
   3. Conditional probabilities and conditional expectation
3. Markov Chains
   1. Discrete-time Markov chains, finite state space: absorbing chains, ergodic chains, stationary distributions, fundamental matrix
   2. Discrete-time Markov chains, countable state space: classification of states and chains, time reversible chains
   3. Stopping times
4. Limit Theorems
   1. Modes of convergence for sequence of random variables
   2. Weak Law of Large Numbers
   3. Strong law of large numbers
   4. Characteristic functions
   5. Central limit theorem
   6. Cramer’s theorem