Answer four core statistics questions

Quick Overview

This multipart problem evaluates competency in probability and mathematical statistics, covering order statistics and covariance, the rationale for degrees of freedom in sample variance, z-test sample size calculation, and a one-sided concentration inequality, categorized under Statistics & Math for a Data Scientist role and combining conceptual derivations with practical formula application. Such questions are commonly asked to verify mathematical rigor in deriving expectations and covariances, understanding sampling distributions and unbiased estimation, applying hypothesis testing power formulas, and proving probabilistic inequalities that underpin statistical reasoning.

Answer four core statistics questions

Company: Two Sigma

Role: Data Scientist

Category: Statistics & Math

Difficulty: easy

Interview Round: Technical Screen

## Problem set (timed) Answer the following four questions. ### 1) Covariance of order statistics Let \(X\) and \(Y\) be independent \(\mathrm{Unif}(0,1)\). Define: - \(U = \min(X, Y)\) - \(V = \max(X, Y)\) Compute \(\mathrm{Cov}(U, V)\). ### 2) Why sample variance uses \(n-1\) Explain why the usual sample variance \[s^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i - \bar x)^2\] uses **degrees of freedom \(n-1\)** instead of \(n\). ### 3) Sample size for a z-test How do you estimate the required sample size for a **z-test** to achieve significance level \(\alpha\) and power \(1-\beta\) for detecting a difference \(\Delta\)? State the formula and assumptions. ### 4) Prove Cantelli’s inequality Prove the one-sided Chebyshev (Cantelli) inequality: If \(X\) has mean \(\mu\) and variance \(\sigma^2\), then for any \(a>0\), \[\Pr(X-\mu \ge a) \le \frac{\sigma^2}{\sigma^2 + a^2}.\]

Quick Answer: This multipart problem evaluates competency in probability and mathematical statistics, covering order statistics and covariance, the rationale for degrees of freedom in sample variance, z-test sample size calculation, and a one-sided concentration inequality, categorized under Statistics & Math for a Data Scientist role and combining conceptual derivations with practical formula application. Such questions are commonly asked to verify mathematical rigor in deriving expectations and covariances, understanding sampling distributions and unbiased estimation, applying hypothesis testing power formulas, and proving probabilistic inequalities that underpin statistical reasoning.

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Dec 15, 2025, 12:00 AM
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Problem set (timed)

Answer the following four questions.

1) Covariance of order statistics

Let XX and YY be independent Unif(0,1)\mathrm{Unif}(0,1). Define:

  • U=min⁡(X,Y)U = \min(X, Y)
  • V=max⁡(X,Y)V = \max(X, Y)

Compute Cov(U,V)\mathrm{Cov}(U, V).

2) Why sample variance uses n−1n-1

Explain why the usual sample variance

s2=1n−1∑i=1n(xi−xˉ)2s^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i - \bar x)^2

uses degrees of freedom n−1n-1 instead of nn.

3) Sample size for a z-test

How do you estimate the required sample size for a z-test to achieve significance level α\alpha and power 1−β1-\beta for detecting a difference Δ\Delta? State the formula and assumptions.

4) Prove Cantelli’s inequality

Prove the one-sided Chebyshev (Cantelli) inequality: If XX has mean μ\mu and variance σ2\sigma^2, then for any a>0a>0,

Pr⁡(X−μ≥a)≤σ2σ2+a2.\Pr(X-\mu \ge a) \le \frac{\sigma^2}{\sigma^2 + a^2}.

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