How do you derive CDF from a PDF?
Company: Uber
Role: Data Scientist
Category: Statistics & Math
Difficulty: easy
Interview Round: Technical Screen
You are given a continuous random variable \(X\) with probability density function (PDF) \(f_X(x)\).
1. Write the definition of the cumulative distribution function (CDF) \(F_X(x)\) in terms of \(f_X(x)\).
2. State the conditions a valid PDF and CDF must satisfy.
3. If \(f_X(x)\) is only defined piecewise (different formulas on different intervals), explain how you would compute \(F_X(x)\) correctly across all ranges of \(x\).
4. (Practical) If \(f_X(x)\) has no closed-form integral, describe a reasonable numerical approach to approximate \(F_X(x)\) for a given \(x\), and how you’d control error.
Quick Answer: This question evaluates understanding of the relationship between a probability density function and its cumulative distribution function, the formal properties that PDFs and CDFs must satisfy, and competence with piecewise definitions and numerical approximation considerations.