Answer the following three interview questions. ## 1) Basic properties of a Markov (transition) matrix Let \(P\) be a transition matrix of a (time-homogeneous) Markov chain on \(n\) states. - State the defining properties of \(P\) (stochasticity, non-negativity, etc.). - State key linear-algebra properties that always hold (e.g., eigenvalue facts, stationary distribution existence conditions). - Briefly describe conditions under which \(P^k\) converges as \(k\to\infty\) and what it converges to. ## 2) OLS properties after a rotation (two independent uniforms) You observe i.i.d. data \((x_{i1}, x_{i2}, y_i)\) for \(i=1,\dots,n\), where - \(x_{i1}, x_{i2}\) are independent and identically distributed \(\mathrm{Unif}(-1,1)\) (mean 0), - the data follow a linear model \(y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \varepsilon_i\) with \(\mathbb{E}[\varepsilon_i\mid x_{i1},x_{i2}]=0\) and \(\mathrm{Var}(\varepsilon_i\mid x_{i1},x_{i2})=\sigma^2\). You fit OLS with an intercept. Now define a rotated feature vector \[x_i^* = R\,x_i,\qquad x_i = (x_{i1},x_{i2})^\top,\] where \(R\) is a 2D rotation matrix (orthonormal: \(R^\top R=I\)). You refit OLS using \(x_i^*\) (with an intercept). - How do the estimated coefficients transform (relationship between \(\hat\beta\) and \(\hat\beta^*\))? - Are the fitted values \(\hat y\) invariant to this rotation? - Under what conditions (on the design distribution) is the *sampling distribution* of the slope estimator “rotation-invariant” (at least in second moments)? ## 3) Two players flipping coins in parallel A and B flip fair coins simultaneously each round. - A’s stopping time \(T_A\): the first round \(t\ge 2\) at which A has flipped **two consecutive heads** (i.e., A has \(H\) at rounds \(t-1\) and \(t\)). - B’s stopping time \(T_B\): the first round \(t\ge 1\) at which B flips a **tail**. Compute \(\mathbb{P}(T_A < T_B)\). (If both events happen in the same round, that is not counted as “A before B”.)