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Answer four core statistics questions

Two SigmaData Scientist · Technical Screen
EasyData ScientistTechnical ScreenStatistics & Math

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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.

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}.]