Assess Fundamental Statistics Knowledge in Data-Science Interviews

Scenario

Interview assessing fundamental statistics knowledge for a data-science role

Question

Given an i.i.d. sample from a Normal distribution, derive the maximum-likelihood estimators for its mean and variance. 2) For random variables X and Y with a known joint pdf, derive the conditional distribution of X given Y. 3) Write the pdf and cdf of the standard Normal distribution. 4) Starting from a concrete estimation task, list and justify the assumptions required for the estimator to be unbiased.

Hints

Use log-likelihood differentiation, f(x|y)=f(x,y)/f_Y(y), Φ(z)=∫_{-∞}^z φ(t)dt, and recall iid sampling and finite expectation conditions for unbiasedness.