This is basically a numerical integration problem.
Clarify the available PDF function's signature and whether it returns the density at a point or can integrate. Then implement the CDF using numerical integration (e.g., trapezoidal rule) or by leveraging symmetry and the error function if the PDF is standard normal. Discuss trade-offs between accuracy and computational cost, and consider edge cases like extreme values.
Pro tip: Mention that for a standard normal, the CDF can be computed via the error function (erf) if available, but since only the PDF is given, numerical integration is necessary. Also, highlight that for extreme tails, direct integration may underflow, so using a transformation or approximation (e.g., Abramowitz and Stegun) might be needed.
Ask whether the PDF is for a standard normal or general normal, and whether it's callable at any point. Confirm if the CDF should be implemented from scratch using only the PDF.
Select a method like trapezoidal rule or Simpson's rule for integrating the PDF from -infinity to x. Discuss step size and accuracy trade-offs.
Use symmetry: CDF(x) = 0.5 + integral from 0 to x for standard normal. For general normal, transform to standard normal. For x < 0, use CDF(x) = 1 - CDF(-x).
Write code that integrates the PDF, handling edge cases like very large |x|. Test against known values (e.g., CDF(0)=0.5, CDF(1.96)≈0.975).
Mention that numerical integration can be slow for many calls; consider precomputing a table or using approximation formulas if performance is critical.
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Structure your answer around the experiment lifecycle: define the hypothesis and target population, specify randomization and metrics, then address confounders and mitigation. Emphasize the ML engineer's role in ensuring valid randomization, reliable metric computation, and detecting subtle biases.
Pro tip: Mention that promotions often have heterogeneous treatment effects—segment by user value or cuisine preference to avoid diluting the average effect. Also, highlight the importance of guardrail metrics like delivery time and courier utilization to catch unintended consequences.
Clarify the promotion's goal (e.g., increase orders) and specify who is eligible (e.g., new users, lapsed users). Define the treatment and control groups.
Choose the randomization unit (user, session, or order) and ensure it's consistent with the metric and avoids contamination. Consider stratified randomization by key covariates like city or user tenure.
Identify primary metrics (e.g., conversion rate, order frequency), secondary metrics (e.g., average order value, retention), and guardrail metrics (e.g., delivery time, courier utilization).
List potential confounders such as seasonality, city-level differences, and novelty effects. Plan to control them via stratification, regression adjustment, or holdout periods.
Use statistical tests (e.g., t-test, CUPED) to measure effects, check for heterogeneous treatment effects, and decide on rollout or further experiments.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.