Assess Fundamental Statistics Knowledge in Data-Science Interviews

Quick Overview

Google data scientist statistics prompt covering MLEs for normal mean and variance, conditional density from a joint pdf, standard normal pdf and cdf, and assumptions for unbiased estimators.

Assess Fundamental Statistics Knowledge in Data-Science Interviews

Company: Google

Role: Data Scientist

Category: Statistics & Math

Difficulty: medium

Interview Round: Technical Screen

##### 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.

Overview: Google data scientist statistics prompt covering MLEs for normal mean and variance, conditional density from a joint pdf, standard normal pdf and cdf, and assumptions for unbiased estimators.

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Jul 12, 2025
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Fundamental Statistics for a Data Science Interview

You are given several standard statistics tasks commonly used in a data-science technical screen.

Constraints & Assumptions

  • For the normal MLE question, assume X_1, ..., X_n are i.i.d. samples from Normal(mu, sigma^2) .
  • For the conditional-density question, assume all densities are well-defined and the marginal density in the denominator is positive.
  • For unbiasedness, start with a concrete estimator and state the assumptions needed for that estimator.
  • Show enough derivation to demonstrate understanding, not just final formulas.

Clarifying Questions to Ask Guidance

  • Should I derive the MLEs fully or give the main log-likelihood steps?
  • Are X and Y continuous, discrete, or mixed?
  • Should the standard normal CDF be written as an integral or using the error function?
  • Which estimator should I use for the unbiasedness example?

What a Strong Answer Covers Guidance

  • Normal MLE derivation: write the likelihood/log-likelihood, differentiate with respect to mu and sigma^2 , and show that mu_hat = sample mean and sigma_hat^2 = (1/n) * sum((x_i - x_bar)^2) .
  • The distinction between MLE variance with denominator n and unbiased sample variance with denominator n - 1 .
  • Conditional density formula f_{X|Y}(x|y) = f_{X,Y}(x,y) / f_Y(y) , where f_Y(y) is obtained by integrating over x .
  • Standard normal pdf and cdf definitions.
  • A concrete unbiasedness example such as the sample mean, with assumptions like finite mean, sampling from the target population, and no systematic measurement or selection bias.
  • A brief verification using linearity of expectation.

Follow-up Questions Guidance

  • Why is the MLE for variance biased?
  • Does independence matter for unbiasedness of the sample mean?
  • How would the conditional formula change for discrete variables?
  • What assumptions affect variance versus bias?

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