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wtw·Software Engineer·Technical Phone Screen·Intermediate

Intermediate
Apr 2026

Summary

Interviewed for a quant engineer role at WTW, got a stats theory question that felt more like a probability exam than anything practical. Not a bad experience, just not what I expected.

Questions Asked (1)

Q1

Given a normally distributed random variable X with mean mu and variance sigma squared, walk through how you standardize it. Derive the result, prove the transformed variable follows a standard normal distribution, and explain why standardization matters in practice.

Technical Trade-offsProduct Analytics & Metrics
Author's notes

I knew the derivation cold so that part was fine.

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AI HintsAI Generated

Suggested Approach

Start by defining the standardization formula Z = (X - μ)/σ, then derive its mean and variance using properties of expectation and variance. Prove normality by using the linear transformation property of normal distributions or moment generating functions, and conclude by discussing practical applications such as comparing scores, hypothesis testing, and machine learning feature scaling.

Pro tip: Emphasize that standardization is a linear transformation, so normality is preserved, and connect it to real-world scenarios like A/B testing or model convergence to show practical insight.

1. Define Standardization

Introduce the standardization formula Z = (X - μ)/σ, explaining that it shifts the mean to 0 and scales the variance to 1.

2. Derive Mean and Variance of Z

Compute E[Z] = 0 and Var(Z) = 1 using linearity of expectation and variance properties: Var(aX + b) = a^2 Var(X).

3. Prove Normality of Z

Use the fact that a linear transformation of a normal variable is normal, or derive the MGF of Z and show it matches the standard normal MGF.

4. Explain Practical Importance

Discuss why standardization matters: enables comparison across different scales, simplifies statistical inference, and is crucial for algorithms like gradient descent and PCA.

Key Points to Mention

  • Standardization formula: Z = (X - μ)/σ
  • Mean of Z is 0, variance of Z is 1
  • Linear transformation preserves normality
  • Moment generating function (MGF) proof: M_Z(t) = e^{t^2/2}
  • Practical uses: comparing scores, hypothesis testing, feature scaling in ML
  • Connection to Z-scores and standard normal table

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