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Past exams till 2020
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- This document provides a comprehensive collection of exercises and solutions from various exams on Stochastic Dynamical Models.
- Discrete-Time Markov Chains (DTMC):
- Focuses on constructing transition matrices and classifying states as recurrent or transient.
- Covers the computation of invariant distributions and mean arrival/sojourn times.
- Includes problems on martingales and applying stopping theorems for expected values.
- Examples include post office queues, bug movement in a square, and gambler's ruin problems.
- Continuous-Time Markov Chains (CTMC):
- Explores the development of transition rate (Q) matrices and transition probability P(t) matrices.
- Addresses irreducibility, invariant densities, and mean arrival/waiting times in continuous settings.
- Martingale properties and stopping theorems are applied to CTMCs.
- Applications range from ATM cash machines and car washing systems to labor market dynamics and article publishing processes.
- Key Concepts and Techniques:
- State Classification: Identifying recurrent, transient, and absorbing states and classes.
- Invariant Distributions: Solving systems of linear equations (e.g., πP = π or πQ = 0) to find stationary probabilities.
- Mean Passage Times: Calculating average times to reach specific states or sets of states, often using linear systems or stopping theorems.
- Martingales: Demonstrating martingale properties for various processes and utilizing them with stopping theorems (e.g., E[M_T] = E[M_0]).
- Characteristic Equations: Employed for solving difference equations arising from mean time computations or invariant distributions.
- Stopping Theorems: A crucial tool for computing expected values at stopping times.
- The problems presented offer diverse scenarios, requiring a solid understanding of Markov chain theory and its analytical methods for predicting long-term behavior and probabilities.
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