Solve Markov and distribution expectation problems

Quick Overview

This interview question evaluates statistical assumptions, formulas, estimation strategy, uncertainty, edge cases, and interpretation in a realistic interview setting. A strong answer for Solve Markov and distribution expectation problems states assumptions, handles edge cases, explains trade-offs, and shows how to validate the result clearly.

Solve Markov and distribution expectation problems

Company: DRW

Role: Data Scientist

Category: Statistics & Math

Difficulty: medium

Interview Round: Onsite

##### Question Given a finite Markov chain, how do you find its stationary (steady-state) distribution? State and derive the expected value of an exponential distribution and of a Poisson distribution. 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. In a specified two-player game-theory scenario, calculate the probability of a particular outcome.

Quick Answer: This interview question evaluates statistical assumptions, formulas, estimation strategy, uncertainty, edge cases, and interpretation in a realistic interview setting. A strong answer for Solve Markov and distribution expectation problems states assumptions, handles edge cases, explains trade-offs, and shows how to validate the result clearly.

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Aug 4, 2025, 10:55 AM
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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):

  • 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.

Clarifying Questions to Ask Guidance

  • Clarify the random variables, distributional assumptions, independence assumptions, and desired output.
  • Show enough derivation for the interviewer to follow the reasoning.
  • Explain how you would validate the result with simulation or sensitivity checks.

What a Strong Answer Covers Guidance

  • A correct setup with definitions, formulas, and boundary conditions.
  • A step-by-step derivation or estimation plan.
  • Interpretation of the result, including uncertainty and practical limitations.
  • Checks for assumptions, edge cases, and numerical stability.

Follow-up Questions Guidance

  • How would the result change if the assumptions were relaxed?
  • Can you verify the answer with a simulation?
  • What is the most likely source of estimation error?
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