← Citadel Interview Insights

Citadel·Software Engineer·Technical Phone Screen·Intermediate

IntermediatePrefer not to say
May 2026

Summary

Citadel quant interview, one probability question that looks straightforward until you realize you need to actually remember what a triangular distribution is. Walked away thinking I got it but honestly wasn't sure until I checked the math later.

Questions Asked (1)

Q1

X and Y are independent uniform random variables on [0, 100]. What is P(50 ≤ |X − Y| ≤ 80)?

Algorithms & Data StructuresTechnical Trade-offs
Author's notes

The key move is recognizing that |X − Y| follows a triangular distribution on [0, 100] with PDF f(d) = 2(100 − d) / 10000.

Create a free account to read the full note

AI HintsAI Generated

Suggested Approach

Model the problem geometrically by considering the unit square [0,100]x[0,100] and identifying the region where 50 ≤ |X−Y| ≤ 80. Compute the area of this region and divide by 10000 to get the probability.

Pro tip: After computing the probability, sanity-check it by considering the extreme cases: P(|X−Y| ≤ 50) should be 0.75 (since the area of the band |x−y| ≤ 50 is 7500), and P(|X−Y| ≤ 80) should be 0.96 (area 9600). The difference 0.21 is plausible.

1. Set up the geometric representation

Draw the 100x100 square representing all possible outcomes (X,Y). The total area is 10000.

2. Identify the region of interest

The condition 50 ≤ |X−Y| ≤ 80 corresponds to the area between the lines y = x ± 50 and y = x ± 80, within the square.

3. Compute the area of the region

Calculate the area of the band where |X−Y| ≤ 80 and subtract the area where |X−Y| ≤ 50. Alternatively, compute the two triangular strips directly.

4. Calculate the probability

Divide the area of the region by the total area (10000) to obtain the probability.

5. Verify and present the answer

Check the result for plausibility (e.g., between 0 and 1) and present the final probability, e.g., 0.21 or 21%.

Key Points to Mention

  • Geometric probability: area of region divided by total area.
  • The region |X−Y| ≤ d forms a band of width d√2 around the diagonal, with area 10000 − (100−d)^2.
  • Compute P(|X−Y| ≤ 80) = 1 − (20/100)^2 = 0.96.
  • Compute P(|X−Y| ≤ 50) = 1 − (50/100)^2 = 0.75.
  • The desired probability is the difference: 0.96 − 0.75 = 0.21.
  • Alternatively, compute the area of the two triangular strips directly: each strip has area 1050, total 2100, probability 0.21.

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