Answer core probability and statistics questions

Work through the interview-style probability and statistics questions below. Give the relevant formulas and a brief explanation for each.

1. Bayes’ rule: Write down Bayes’ rule. Suppose the disease prevalence is $P(D)=1\%$, the test sensitivity is $P(+\mid D)=0.99$, and the false positive rate is $P(+\mid\neg D)=0.05$; find $P(D\mid +)$.

2. Controls in regression: In observational data, why can including control variables alter the estimated coefficient for the variable you care about? Under what circumstances can adding controls create bias?

3. CLT: State the Central Limit Theorem and explain what it implies in practice about the sampling distribution of the sample mean.

4. Uniform distribution moments: For $X\sim \mathrm{Unif}(a,b)$, calculate $E[X]$ and $\mathrm{Var}(X)$.

5. Hypothesis test / t-stat: When testing $H_0: \mu=\mu_0$ using a sample mean $\bar x$, sample standard deviation $s$, and sample size $n$, write down the one-sample t-statistic.

6. Effect size vs MDE: Define effect size and Minimum Detectable Effect (MDE). How are power, variance, sample size, and alpha related to MDE?

Overview: This question tests fundamental probability and statistical inference skills—including Bayesian reasoning, regression controls and confounding, the Central Limit Theorem, distribution moments, t-statistics, and effect size/MDE—that are relevant to quantitative data analysis and hypothesis testing for a data scientist role in the statistics and math area.

This question came from a data scientist interview experience.