Answer four core statistics questions

Problem set (timed)

Work through the four questions below.

1) Covariance of order statistics

Take $X$ and $Y$ to be independent $\mathrm{Unif}(0,1)$ random variables. 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

$$s^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i - \bar x)^2$$

divides by degrees of freedom $n-1$ rather than $n$.

3) Sample size for a z-test

How would you determine the required sample size for a z-test to reach significance level $\alpha$ and power $1-\beta$ when detecting a difference $\Delta$? Give the formula and its assumptions.

4) Prove Cantelli’s inequality

Prove the one-sided Chebyshev bound known as Cantelli's inequality:

If $X$ has mean $\mu$ and variance $\sigma^2$, then for any $a>0$,

$$\Pr(X-\mu \ge a) \le \frac{\sigma^2}{\sigma^2 + a^2}.$$

Overview:

This four-part problem assesses skills in probability and mathematical statistics: it covers covariance of order statistics, the rationale for degrees of freedom in sample variance, z-test sample size calculation, and a one-sided concentration inequality, and it is classified under Statistics & Math for a Data Scientist role, combining conceptual derivations with hands-on formula use.

These questions are often asked to check mathematical rigor in computing expectations and covariances, understanding sampling distributions and unbiased estimation, applying hypothesis-testing power formulas, and proving probability inequalities that support statistical inference.

Community answers

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