Solve probability and stopping questions
Company: Citadel
Role: Data Scientist
Category: Statistics & Math
Difficulty: hard
Interview Round: Onsite
Answer the following independent interview questions:
1. Let X and Y be independent random variables, each distributed Uniform(0,1). Find the probability density function of S = X + Y.
2. A fair six-sided die is rolled repeatedly until all six faces have appeared at least once. What is the expected number of rolls?
3. A point is sampled uniformly from the volume of the 3D unit ball
{(x, y, z) : x^2 + y^2 + z^2 <= 1}.
What is Var(X)?
4. Online selection problem: you observe n i.i.d. draws from Uniform(0,1) one at a time. After seeing each draw, you must immediately decide whether to keep it or discard it. You may keep at most k draws, and your goal is to maximize the expected sum of the kept values. What is the optimal strategy, and how can the optimal expected value be computed?
Quick Answer: This set of problems evaluates probabilistic reasoning and mathematical statistics skills, covering convolution for sums of independent variables, expectation of stopping times, variance under geometric sampling in continuous volumes, and online selection/optimal stopping for maximizing accumulated reward.