Solve Markov and distribution expectation problems
Overview
This question evaluates understanding of stochastic processes and stationary distributions (finite Markov chains), expectations and moment calculations for common probability distributions (Exponential and Poisson), spectral relationships in linear algebra (eigenvalues of matrix inverses), and probabilistic outcome computation in mixed-strategy games. It is commonly asked to assess foundational competence in Statistics & Math—covering probability, linear algebra, and game theory—and requires both conceptual understanding of theoretical relationships and practical application of formulas within these domains.
Statistics, Linear Algebra, and Game Theory Fundamentals
1) Stationary Distribution of a Finite Markov Chain
Given a finite Markov chain with transition matrix P, how do you compute its stationary (steady-state) distribution?
2) Expectations of Common Distributions
(a) State and derive the expected value of an Exponential(λ) distribution.
(b) State and derive the expected value of a Poisson(λ) distribution.
3) Eigenvalues of an Inverse Matrix
For a non-singular square matrix A, express the sum of the eigenvalues of A⁻¹ in terms of the eigenvalues of A, and explain why.
4) Probability of an Outcome in a Two-Player Game
Assume two players choose actions independently according to mixed strategies. For a 2×2 game where Player 1 plays Top with probability p (Bottom with 1−p) and Player 2 plays Left with probability q (Right with 1−q):
- Calculate the probability of the outcome (Top, Left).
- Generalize your expression to an m×n game where Player 1 uses probabilities (p₁, …, p_m) and Player 2 uses (q₁, …, q_n) over their respective actions.
Solution
Login required