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Bank of America·Software Engineer·Technical Phone Screen·Intermediate

IntermediatePrefer not to say
Jun 2026

Summary

Quant Engineer interview at Bank of America that leaned heavily into statistics fundamentals. The probability and estimation questions were more rigorous than I expected for what I thought would be a more coding-focused screen.

Questions Asked (1)

Q1

Define the mean and variance of a random variable, then for a sample x_1 through x_n: is the sample mean an unbiased estimator? Is the sample variance biased when you divide by n, and why does dividing by (n-1) fix that? Walk through the algebra.

Technical Trade-offsA/B Testing & Experimentation
Author's notes

This looked like a warmup and turned into a 15-minute deep dive.

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AI HintsAI Generated

Suggested Approach

Start by clearly defining the mean and variance of a random variable, then transition to the sample estimators. Explain unbiasedness of the sample mean, then show the algebra for why the sample variance with n is biased and how (n-1) corrects it, emphasizing the loss of a degree of freedom due to estimating the mean.

Pro tip: Mention that the bias arises because deviations are taken from the sample mean, which is itself estimated from the data, leading to underestimated variability. This shows deep understanding and connects to practical implications in A/B testing where unbiased variance estimates are crucial.

1. Define population parameters

Define the mean (expected value) and variance of a random variable X: μ = E[X], σ² = E[(X-μ)²] = E[X²] - (E[X])².

2. Define sample estimators

Define the sample mean x̄ = (1/n)Σx_i and sample variance with denominator n: s²_n = (1/n)Σ(x_i - x̄)².

3. Show sample mean is unbiased

Compute E[x̄] = (1/n)ΣE[x_i] = μ, proving unbiasedness.

4. Derive bias of sample variance with n

Show E[s²_n] = σ² - σ²/n = σ²(n-1)/n, hence biased downward. Use algebra: Σ(x_i - x̄)² = Σ(x_i - μ)² - n(x̄ - μ)², take expectations.

5. Explain correction with (n-1)

Define s² = (1/(n-1))Σ(x_i - x̄)². Since E[s²] = σ², it's unbiased. Explain that dividing by n-1 corrects for the fact that deviations are from the sample mean, which underestimates variability.

Key Points to Mention

  • Definition of expected value and variance for a random variable.
  • Sample mean is an unbiased estimator: E[x̄] = μ.
  • Sample variance with denominator n is biased: E[s²_n] = σ²(n-1)/n.
  • Algebraic identity: Σ(x_i - x̄)² = Σ(x_i - μ)² - n(x̄ - μ)².
  • Degrees of freedom: estimating the mean consumes one degree of freedom, so divide by n-1.
  • Unbiased sample variance: E[s²] = σ² when using n-1.

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.