Answer the following probability/statistics sub-questions: 1) Given a finite-state, irreducible, aperiodic Markov chain with transition matrix P (provided), compute its stationary distribution π by solving πP = π with ∑i πi = 1, and state why the solution is unique. 2) Let X ~ Exponential(λ) and N ~ Poisson(λ), with λ > 0. Derive E[X] and E[N] from first principles (integration/summation), showing intermediate steps. 3) For a 2×2 zero-sum game with payoff matrix to Player A given in the prompt, find the mixed-strategy Nash equilibrium and report the probability each player assigns to their first action (use probability calculations to justify the equilibrium).

Compute Markov steady state and expectations evaluates statistical assumptions, formulas, estimation strategy, uncertainty, edge cases, and interpretation in a realistic interview setting. A strong answer states assumptions, handles edge cases, explains trade-offs, and shows how to validate the result clearly.

How do I approach Statistics & Math interview questions?

Statistics & Math questions require understanding of core concepts and practice. PracHub provides solutions with explanations to help you master statistics & math interviews.

What difficulty level is this interview question?

This is a medium difficulty Statistics & Math question, commonly asked during Onsite rounds at DRW.

What role is this question designed for?

This question is commonly asked for Data Scientist candidates at DRW during technical interviews.

Compute Markov steady state and expectations

Probability and Game Theory: Three Sub-questions

Context: The exact transition matrix P (for Q1) and the 2×2 payoff matrix (for Q3) are not provided. Below, you will (a) solve generically in symbolic form, and (b) see a small numeric example to illustrate the procedure.

Finite-state Markov chain

Given a finite-state, irreducible, aperiodic Markov chain with transition matrix P, compute its stationary distribution π by solving πP = π with ∑ᵢ πᵢ = 1. Explain why the solution is unique.

Expectations from first principles

Let X ~ Exponential(λ) and N ~ Poisson(λ) with λ > 0. Derive E[X] and E[N] from first principles (integration/summation), showing intermediate steps.

2×2 zero-sum game

For a 2×2 zero-sum game with Player A’s payoff matrix [ [a, b], [c, d] ], find the mixed-strategy Nash equilibrium. Report the probability each player assigns to their first action (row 1 for A, column 1 for B), and justify the equilibrium with probability calculations.

Constraints & Assumptions

Preserve the scope, facts, inputs, and requested outputs from the prompt above.
If the prompt leaves a detail unspecified, state a reasonable assumption before relying on it.
Keep the answer interview-ready: concise enough to present, but concrete enough to implement or evaluate.

Clarifying Questions to Ask

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

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

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?