1. Random events and Probability: counting, operations, probability space, urn problem, etc
2. Conditional Probability: conditioning, independence, formulae, Baysian, etc.
3. Random Variables and Distributions : rv and distribution function, discrete-type rv , continuous-type rv, joint (multivariate)- type rv, sum and max of i.i.d. rv’s, conditional-type rv, etc
4. Expectation, Variance, and other Macro-values: expected value, formulae, variance and covariance, correlation, conditional expectation, moment generating function, etc
5. Limits Theorems: some inequalities (Markov, Chebyshev, Chernoff, Cauchy-Schwarz), law of large numbers, DeMoirve-Laplace Theorem, Central Limit Theorem, Poisson Limit Theorem, Approximation of binomial distribution, etc.
6. Poisson Process: inter-arrival and waiting times, Poisson v.s. exponential distributions, compound Poisson process(optional)
7. Finite-state Markov Chain: Random walk, Markov property, Markov matrix, Chapman-Kolmogorov equation, state-classification, invariant distribution, continuous-time MC ( the last three are optional)