Sample uniformly from a circle’s area
Company: LinkedIn
Role: Machine Learning Engineer
Category: Coding & Algorithms
Difficulty: medium
Interview Round: Onsite
How would you generate a point `(x, y)` uniformly at random from the *area* of a circle of radius `R` centered at the origin?
- Explain why naive choices (e.g., choosing radius uniformly in `[0,R]`) are not uniform over area.
- Provide a correct sampling method (math + pseudocode).
Overview: This question evaluates understanding of continuous probability distributions and geometric random sampling, testing whether a candidate recognizes how area-weighted uniformity differs from naive radial choices.
Read the full LinkedIn Machine Learning Engineer interview experience this question came from
Community answers
Answer by Cuckoo
I first choose a random angle uniformly from 0 to 2π. I cannot choose the radius uniformly because outer rings have more area. I use r = R * sqrt(U) to compensate for the increasing area as the radius gets larger. Then I convert from polar coordinates to x and y.
r = R * sqrt(random())
theta = 2 pi random()
x = r * cos(theta)
y = r * sin(theta)
Answer by Cuckoo
Another solution is Rejection sampling. Suppose we have a square around the circle whose length of each side is 2R. Now, we generate numbers between 0 and 2R for x an y and every point has the same chance to get selected. Then if the selected point is inside the circle, x2 + y**2 < R, we accept it otherwise generate another number.
The chance of success in this experiment is piR2/4R**2