Knew the answer conceptually but fumbled the formal setup under pressure.
Start by defining the stationary distribution as a probability vector π satisfying πP = π, then explain how to solve this as a left eigenvector problem or linear system. Emphasize the conditions for existence and uniqueness (irreducibility and positive recurrence) and mention computational methods for large chains.
Pro tip: In practice, for large state spaces, direct linear algebra is infeasible; power iteration or Markov Chain Monte Carlo (MCMC) methods are preferred. Also, always check that the chain is irreducible and aperiodic to guarantee a unique stationary distribution.
State that a stationary distribution π is a probability vector (sums to 1, non-negative) such that π = πP, where P is the transition matrix.
Mention that for a finite irreducible chain, a unique stationary distribution exists. If the chain is periodic, the stationary distribution still exists but convergence may not occur.
Set up the system π(P - I) = 0 with the constraint sum(π) = 1. Solve using linear algebra (e.g., Gaussian elimination, eigen-decomposition).
For large chains, use iterative methods like power iteration (π_{k+1} = π_k P) until convergence, or solve the system using sparse linear solvers.
If the chain is reversible, solve the detailed balance equations π_i P_{ij} = π_j P_{ji} for all i,j, which can simplify the computation.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
The exponential part was fine, integration by parts and you get 1/lambda.
Start by clearly stating the expected values: for exponential distribution, E[X] = 1/λ (or 1/β depending on parameterization); for Poisson distribution, E[X] = λ. Then derive each using the definitions: for exponential, integrate x * f(x) from 0 to infinity; for Poisson, sum k * P(X=k) from k=0 to infinity. Show the key steps and mention the memoryless property and Poisson process connection.
Pro tip: Be explicit about the parameterization you're using (rate λ vs. scale β) to avoid confusion, and relate the two distributions via the Poisson process—this shows deeper understanding and is highly relevant for a data science role at a trading firm.
Clearly state that for an exponential distribution with rate λ, E[X] = 1/λ; for a Poisson distribution with rate λ, E[X] = λ. Mention that parameterization matters (e.g., scale θ = 1/λ gives E[X] = θ).
Write the PDF f(x) = λe^{-λx} for x ≥ 0. Compute E[X] = ∫_0^∞ x λe^{-λx} dx using integration by parts (or recognize it as the mean of a gamma distribution with shape 2). Show the steps: let u = x, dv = λe^{-λx} dx, etc., leading to 1/λ.
Write the PMF P(X=k) = e^{-λ} λ^k / k!. Compute E[X] = Σ_{k=0}^∞ k * e^{-λ} λ^k / k!. Simplify by canceling k and shifting the index to get λ e^{-λ} Σ_{j=0}^∞ λ^j / j! = λ e^{-λ} e^{λ} = λ.
Explain that in a Poisson process with rate λ, the number of events in a unit interval is Poisson(λ) and the inter-arrival times are Exponential(λ). This connection reinforces both derivations and highlights practical relevance.
Summarize the results and note key properties: exponential is memoryless, Poisson has variance equal to mean, and both are fundamental in modeling waiting times and event counts.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by recalling that if λ is an eigenvalue of A, then 1/λ is an eigenvalue of A⁻¹, assuming A is invertible. Then the sum of eigenvalues of A⁻¹ is the sum of reciprocals of eigenvalues of A, which can be expressed as the trace of A⁻¹. Explain the reasoning using the eigenvalue definition and properties of inverses.
Pro tip: Mention that the sum of eigenvalues equals the trace, and for A⁻¹ it's the trace of A⁻¹, which is also the sum of reciprocals of eigenvalues of A. This shows you understand the connection between eigenvalues, trace, and matrix inverse.
State that for a non-singular matrix A, if Av = λv for a non-zero vector v, then λ is an eigenvalue of A.
Apply A⁻¹ to both sides of Av = λv to get v = λ A⁻¹ v, which implies A⁻¹ v = (1/λ) v. Thus, 1/λ is an eigenvalue of A⁻¹.
The sum of eigenvalues of A⁻¹ is the sum over all eigenvalues λ_i of A of 1/λ_i. This is also equal to the trace of A⁻¹.
Conclude that if the eigenvalues of A are λ₁, λ₂, ..., λₙ, then the sum of eigenvalues of A⁻¹ is ∑_{i=1}^n 1/λ_i.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the specific two-player game and the outcome in question, as the probability depends on the game's structure and strategies. Then, identify the appropriate solution concept (e.g., Nash equilibrium, mixed strategies) and compute the probability using probability theory and game-theoretic tools. Finally, validate the result with a simple simulation or edge cases.
Pro tip: At DRW, interviewers value clear assumptions and practical validation. State your assumptions explicitly and mention that you would verify the result with a quick Monte Carlo simulation, showing both theoretical rigor and practical implementation skills.
Ask questions to understand the game rules, players' strategies, and the exact outcome whose probability is requested. Confirm whether strategies are pure or mixed, and if the game is simultaneous or sequential.
Determine the appropriate equilibrium concept (e.g., Nash equilibrium, dominant strategies) and whether the probability is under equilibrium play or a specific strategy profile.
Use probability theory and game-theoretic calculations to derive the probability. For mixed strategies, set up equations for indifference conditions and solve for probabilities.
Check the result with a simple simulation or by testing edge cases. Interpret the probability in the context of the game and discuss any implications.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.