I knew the basics: non-negative entries, rows sum to one, eigenvalue of 1 always exists.
Begin by precisely defining the structural properties of a Markov transition matrix, then systematically build up to the spectral theory (eigenvalues), connect eigenvalues to stationary distributions, and finally discuss convergence of matrix powers. Ground abstract concepts with brief intuitive examples to demonstrate both mathematical rigor and applied understanding.
Pro tip: Voleon is a quantitative hedge fund that heavily uses probabilistic and statistical models, so explicitly connecting Markov chain convergence to practical concerns like mixing times, ergodicity in financial simulations, or MCMC sampling will signal that you understand why this theory matters in production settings.
State that a Markov transition matrix P is a square, non-negative matrix where every row sums to 1 (row-stochastic). Clarify that entry P_ij represents the probability of transitioning from state i to state j.
Explain that all eigenvalues λ satisfy |λ| ≤ 1, and λ = 1 is always an eigenvalue (since P·1 = 1, the all-ones vector is a right eigenvector). Mention that for irreducible aperiodic chains, 1 is the unique eigenvalue of maximum modulus, while periodic chains have additional eigenvalues on the unit circle.
Describe the stationary distribution π as the left eigenvector corresponding to eigenvalue 1, satisfying πP = π with π_i ≥ 0 and Σπ_i = 1. Note that for irreducible chains, this stationary distribution is unique (Perron-Frobenius theorem), while reducible chains may have multiple stationary distributions.
State that for an irreducible, aperiodic (ergodic) chain, P^k converges as k→∞ to a matrix where every row equals π. Explain that the rate of convergence is governed by the second-largest eigenvalue modulus |λ_2|, with the spectral gap (1 - |λ_2|) determining the mixing time.
Address what breaks convergence: periodicity (eigenvalues like -1 cause oscillation) and reducibility (absorbing states or multiple communicating classes). Briefly connect to applications such as PageRank, MCMC, or hidden Markov models to demonstrate applied relevance.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Break the question into three distinct sub-problems: (1) how OLS coefficients transform under an orthonormal rotation, (2) whether fitted values (and hence predictions) are preserved, and (3) the distributional invariance condition on the second-moment structure of the features. Ground each answer in the linear algebra of OLS and connect to statistical theory to show depth.
Pro tip: Voleon is a quantitative hedge fund that prizes rigorous mathematical reasoning — explicitly writing out the hat matrix H = X(X'X)⁻¹X' and showing it is invariant under orthonormal column transformations will signal the kind of first-principles thinking they value far more than a verbal hand-wave.
Let X be the n×2 feature matrix and Q a 2×2 orthonormal rotation matrix (Q'Q = I). Define the rotated design matrix X̃ = XQ. Clarify that the intercept column is kept separate and untouched, so the rotation only applies to the two slope columns.
Show that the OLS estimator in the rotated space is β̃ = (X̃'X̃)⁻¹X̃'y = Q'β, where β are the original slope coefficients. This means the coefficient vector rotates by Q' — the individual slope values change, but in a fully determined, invertible way.
The hat matrix H = X̃(X̃'X̃)⁻¹X̃' = XQQ'(QX'XQ')⁻¹Q'X' = X(X'X)⁻¹X', which is identical to the original hat matrix. Therefore ŷ = Hy is unchanged — fitted values and residuals are fully preserved under orthonormal rotation.
Under standard OLS assumptions, Var(β̃) = σ²(X̃'X̃)⁻¹ = σ²Q'(X'X)⁻¹Q. For the sampling distribution of the slope estimator to be invariant in second moments to the choice of Q, we need Q'(X'X)⁻¹Q = (X'X)⁻¹, which holds when X'X ∝ I — i.e., the feature columns are orthogonal and have equal variance (spherical second moments).
The required condition is that the population second-moment matrix Σ_X = E[xx'] is proportional to the identity (isotropic / spherically symmetric features). Under this condition, any orthonormal rotation leaves the covariance structure of β̃ unchanged, making the sampling distribution rotation-invariant in second moments.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Model the joint stopping times of A and B using Markov chains or state-based probability analysis, carefully tracking A's consecutive-heads progress alongside B's single-flip condition. Compute the probability that A's stopping round is strictly less than B's by enumerating joint state transitions and summing over all valid round counts. Verify with a simulation or recursive formula to confirm the closed-form answer.
Pro tip: At a quant-focused firm like Voleon, explicitly setting up the Markov state space and writing out the transition matrix signals rigorous probabilistic thinking — don't just jump to simulation; show you can derive the exact answer analytically before offering simulation as a sanity check.
Clarify that A stops at the first round where the current flip AND the previous flip are both heads (two consecutive heads), while B stops the moment they flip a tail. Confirm both players flip independently each round.
Define A's states: S0 (start or last flip was tails), S1 (last flip was heads, no consecutive yet), and S_done (two consecutive heads seen). Write out transition probabilities: from S0, flip H (prob 1/2) → S1, flip T → S0; from S1, flip H → S_done, flip T → S0.
B stops at round k with probability (1/2)^k, since B needs k-1 heads followed by one tail (geometric distribution with p=1/2). Note B's stopping time T_B ~ Geometric(1/2), so P(T_B = k) = (1/2)^k for k ≥ 1.
For each round k, compute P(A finishes at round k) using the Markov chain (track probability mass in S_done at exactly round k), then multiply by P(T_B > k) = P(B has flipped all heads through round k) = (1/2)^k. Sum over all k ≥ 2 to get P(T_A < T_B).
Use the Markov transition equations to find P(A done at round k) in closed form, sum the series, and simplify. Cross-validate with a Monte Carlo simulation or recursive probability equations to confirm the exact probability (approximately 1/8 or derive precisely as 1/8).
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.