This is a second course in Probability, studying stochastic/random process, intended for majors in Applied Math, Statistics and related fields. The prerequisite is STAT 515 or similar upper-division course. If you did not get at least a B in that course then you will find this course very tough. Moreover, students are required to have some knowledge of Python, as we will cover simulation topics, including sampling of probability distributions, Monte Carlo algorithms, etc. A major focus of the course is on solving problems extending the scope of the lectures, developing analytical skills and probabilistic intuition. The course will cover the following topics in the core of the theory of Random and Stochastic Processes.

1. Review of probability theory : probability space, random variables, expectations, independence, conditional expectations.

2. Random walks and finite state Markov chains: Transition matrix, transience and recurrence, limiting distributions, convergence of Markov Chains.

3. Poisson processes: definition, inter-arrival and waiting time.

4. Continuous Markov chains: Strong Markov properties, Chapman Kolmogorov equations, irreducible and recurrence. Long time behaviour.

5. Brownian Motions: Definitions, scaled random walk, Brownian motion.

In addition, we also add a parallel part to the theoretical lectures - the stochastic simulations.

1.Sampling of basic probability distributions, generation of pseudorandom numbers,

2. Monte Carlo integration Simulation of random samples from discrete distributions and continuous distributions

3. Discrete event simulation for stochastic models of queueing systems

4. MCMC

## Department of Mathematics and Statistics