Solve Markov, OLS-rotation, and coin-toss probability
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 $P$ (e.g., stochasticity, non-negativity).
- 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 HH 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”.)
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Solve Markov, OLS-rotation, and coin-toss probability
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 $P$ (e.g., stochasticity, non-negativity).
- 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 HH 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”.)