##### 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.

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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What difficulty level is this interview question?

This is a medium difficulty Statistics & Math question, commonly asked during Technical Screen rounds at Google.

What role is this question designed for?

This question is commonly asked for Data Scientist candidates at Google during technical interviews.

Assess Fundamental Statistics Knowledge in Data-Science Interviews

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

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

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

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?

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

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

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

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?

Assess Fundamental Statistics Knowledge in Data-Science Interviews

Quick Overview

Assess Fundamental Statistics Knowledge in Data-Science Interviews

Fundamental Statistics for a Data Science Interview

Constraints & Assumptions

Clarifying Questions to Ask

What a Strong Answer Covers

Follow-up Questions

Write your answer

Assess Fundamental Statistics Knowledge in Data-Science Interviews

Quick Overview

Assess Fundamental Statistics Knowledge in Data-Science Interviews

Fundamental Statistics for a Data Science Interview

Constraints & Assumptions

Clarifying Questions to Ask

What a Strong Answer Covers

Follow-up Questions

Write your answer