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

Intermediate
Jun 2026

Summary

Quant Engineer interview at Bank of America with a probability/statistics question that sounds deceptively simple but has real depth to it. The kind of thing where you think you know the answer and then realize you're about to embarrass yourself.

Questions Asked (1)

Q1

X and Y are normally distributed random variables with zero correlation. Does that mean they're independent? Walk through your reasoning.

Technical Trade-offsAlgorithms & Data Structures
Author's notes

My first instinct was to say yes and move on.

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AI HintsAI Generated

Suggested Approach

Start by clarifying that zero correlation does not imply independence in general, but for jointly normal variables it does. Then explain the distinction between uncorrelated and independent, and provide a counterexample for non-normal variables. Conclude with the special case of bivariate normality.

Pro tip: Mention that in finance, returns are often assumed normal, so zero correlation implies independence, but in practice, fat tails and nonlinear dependencies can break this assumption. This shows awareness of real-world model risk.

1. Define the concepts

Define zero correlation (linear relationship) and independence (no relationship whatsoever). Emphasize that independence implies zero correlation, but not vice versa.

2. General case

State that in general, zero correlation does not imply independence. Provide a counterexample, such as Y = X^2 with X symmetric around zero, where correlation is zero but Y depends on X.

3. Special case: joint normality

Explain that if X and Y are jointly normally distributed (bivariate normal), then zero correlation does imply independence. This is because the joint density factorizes when the correlation parameter is zero.

4. Address the question's wording

Clarify that the question says 'normally distributed' but not necessarily 'jointly normal'. If only marginal normality is assumed, zero correlation does not guarantee independence. Provide an example where marginals are normal but joint is not, and correlation is zero yet dependent.

5. Conclusion

Summarize: Under joint normality, zero correlation implies independence; otherwise, it does not. In practice, always check the joint distribution assumption.

Key Points to Mention

  • Independence implies zero correlation, but zero correlation does not imply independence in general.
  • Counterexample: Y = X^2 with X ~ N(0,1) gives Cov(X,Y)=0 but Y is dependent on X.
  • For jointly normal (bivariate normal) variables, zero correlation is equivalent to independence.
  • Marginal normality does not guarantee joint normality; a copula can create dependence with normal marginals.
  • In finance, the normal assumption is common but risky; dependence can exist without correlation.
  • The distinction is crucial in risk management and portfolio theory.

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