Solve Markov and distribution expectation problems
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):
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Calculate the probability of the outcome (Top, Left).
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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.
Constraints & Assumptions
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Preserve the scope, facts, inputs, and requested outputs from the prompt above.
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If the prompt leaves a detail unspecified, state a reasonable assumption before relying on it.
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Keep the answer interview-ready: concise enough to present, but concrete enough to implement or evaluate.
Clarifying Questions to Ask
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Clarify the random variables, distributional assumptions, independence assumptions, and desired output.
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Show enough derivation for the interviewer to follow the reasoning.
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Explain how you would validate the result with simulation or sensitivity checks.
What a Strong Answer Covers
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A correct setup with definitions, formulas, and boundary conditions.
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A step-by-step derivation or estimation plan.
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Interpretation of the result, including uncertainty and practical limitations.
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Checks for assumptions, edge cases, and numerical stability.
Follow-up Questions
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How would the result change if the assumptions were relaxed?
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Can you verify the answer with a simulation?
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What is the most likely source of estimation error?