Stochastic Processes
Homework Solutions for Stochastic Processes Course as Computer Science B.Sc. Student at Department of Mathematical Sciences, Sharif University of Technology
Homework Solutions for Stochastic Processes Course as Computer Science B.Sc. Student at Department of Mathematical Sciences, Sharif University of Technology
Spring 2024
Supervisor: Dr. Hossein Peyvandi
Description
This repository includes my homework and projects around Stochastic Processes.
Homework Assignments
The homeworks folder contains a series of assignments designed to test your knowledge and application of stochastic processes.
| Homework Title | Keywords |
|---|---|
| Homework 1 | gambler, probability, ruin, success, recursive equation, expected steps |
| Homework 2 | Ehrenfest chain, transition matrix, stationary distribution |
| Task 1 | voter model, Markov chain, stable distributions, random walk, return probability |
| Task 2 | complete graph, expected hitting time, vertex, Markov chain, transition matrix, stationary distributions, grid recoloring, absorbing state, particle splitting, branching process, symmetric random walk, return time, expected returns, sequence observation, coin flips, game theory |
| Task 3 | dice rolls, Poisson process, negative binomial distribution, gamma distribution, interarrival times, intensity function, container, Markov chain, empty time, Yule process, transition probability, expected total age, Poisson-headed demon, expected strikes, defeat probability |
| Task 4 | single-server queue, Poisson distribution, service time distribution, queue dynamics, taxi stand, passenger queue, Markov chain, balance equations, stationary distribution, grid color dynamics, absorbing states |
Project
The purpose of this project is to provide an in-depth analysis and application of various stochastic processes. It explores mathematical models and their real-world applications, highlighting key concepts and theories in stochastic processes. In this notebook, we delve into various fundamental and advanced concepts that underpin stochastic processes. We cover Gaussian distributions, Gaussian processes, Brownian motion, and Markov Chain Monte Carlo (MCMC) methods, providing practical implementations and insights into each area.
What You Will Learn and Do
- Gaussian Distributions: You’ll start with univariate and multivariate Gaussian distributions, learning how to fit these models to data and interpret the results.
- Gaussian Processes: You will generate Gaussian processes using specific mean functions and kernels, visualizing different realizations to understand their variability.
- Brownian Motion: Explore the characteristics of Brownian motion, simulate its trajectories, and apply detrending methods to analyze its behavior.
- Markov Chain Monte Carlo (MCMC): Dive into MCMC, starting with a simple toy dataset and progressing to complex models. Learn about the Metropolis-Hastings algorithm, understand how to set up and visualize MCMC chains, and extend your knowledge to fit quadratic models.
Feedback
Your feedback is highly appreciated! If you have any comments or find any discrepancies in the answers, please feel free to open an issue or submit a pull request. Collaboration and discussion are encouraged as it helps all of us learn and grow.