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

Solve Markov, OLS-rotation, and coin-toss probability

Quick Overview

This multi-part question evaluates understanding of Markov chain and stochastic matrix properties, linear-algebra facts about eigenvalues and stationary distributions, invariance and sampling-distribution properties of OLS under orthonormal rotations, and discrete stopping-time probability reasoning for coin-flip processes.

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”.)

Quick Overview

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”.)

Solve Markov, OLS-rotation, and coin-toss probability

Quick Overview

1) Basic properties of a Markov (transition) matrix

2) OLS properties after a rotation (two independent uniforms)

3) Two players flipping coins in parallel

Solution

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Solve Markov, OLS-rotation, and coin-toss probability

Quick Overview

1) Basic properties of a Markov (transition) matrix

2) OLS properties after a rotation (two independent uniforms)

3) Two players flipping coins in parallel

Solution

Comments (0)