Random Variables Interview Questions
This topic is the working language of every quant desk: how a random variable is distributed, how it transforms, and how it co-moves with others. You'll master single-variable tools (PDFs/CDFs, moments, MGFs/PGFs, change-of-variables) and the multivariate machinery that actually drives finance — joi
How to think about random variables questions
A random variable is just a number that hasn't happened yet, and almost every problem here is about pushing that uncertainty through a transformation and asking what comes out the other side. Master the handful of moves for reshaping a distribution and the rest is substitution.
TRANSFORM CLEANLY
When you apply a function to a random variable, the density gets stretched by the local change of scale — the Jacobian. The cleaner route is often the CDF method: chase the event “g(X) ≤ t” back to a statement about X you can already evaluate, then differentiate.
LET MOMENTS DO THE WORK
You rarely need the full distribution. The moment generating function packages every moment into one object and turns sums of independent variables into products — which is exactly why the same few distributions (normal, Poisson, exponential) keep reappearing as the answer.
The thread: identify the distribution, push it through the transformation, and read off mean and variance — you'll meet the same five distributions wearing different costumes.
Random Variables questions (38)
- Variance of a Sum of Independent Random Variables
- Radial Distance Distribution in a Uniform Disk
- Exponent Sum from Symmetric Polynomials
- Correlation Under Linear Transformations
- PGF of a Binomial Random Variable
- Distribution of a Linear Combination of Independent Normals
- Variance and Expectation Inequality
- Log-Likelihood of the Multivariate Normal Distribution
- Generating Correlated Normal Random Variables
- Variance of a Coordinate on the Unit Sphere
- Dependent but Uncorrelated Random Variables
- Correlation of Sum and Difference
- Correlation Between a Variable and Its Noisy Copy
- Correlation of a Sum
- Generating Normal Random Variables from Uniforms
- Order Statistics of Exponentials and Spacing Structure
- Zero Correlation Does Not Imply Independence
- PDF of a Sum of Uniform Random Variables
- Even Moments from an MGF via Taylor Series
- Simulating Equicorrelated Normals via a Factor Model
- Uniform Sampling from the 3D Unit Ball
- Product of Independent Lognormals
- Range of Standard Deviation for a Sum of Correlated Normals
- PDF of the Maximum of Uniform Random Variables
- Sampling a 2D Distribution from 1D Generators
- Normal Mixtures and Products: When Marginal Normality Does Not Imply Joint Normality
- CDF of the X-Projection From a Uniform Point on a Circle
- Length of a Random Decreasing Run from Uniform Draws
- Winsorized Normal Variable Moments
- Joint PDF of the Max and Min of Uniform Random Variables
- Uniform Order Statistics: Mean and Variance
- PDF of a Product of Uniform Random Variables
- Kurtosis of a Gaussian Mixture Return Distribution
- Most Likely Split of Two Normals Given Their Sum
- Uniform Point in the Unit Disk Without Rejection
- Correlation Between Consecutive Streak Lengths
- Joint Density and Spread of Uniform Order Statistics
- Distribution of a Product Raised to a Uniform Power
Random Variables interview questions FAQ
What kind of random variables questions show up in quant interviews?
This page collects 38 random variables problems that recur in quant trading and research interviews, each with a full worked solution and the intuition behind it. They range from quick warmups to the harder variants firms use to separate candidates.
How hard are random variables interview questions?
The set spans 12 easy, 21 medium and 5 hard problems. Most sit at medium difficulty — a few minutes of clean reasoning — with a harder tail that rewards knowing the canonical approach rather than grinding.
How should I practice random variables for quant interviews?
Work through them by difficulty, starting just below your level, and write the solution out before checking. 4 are free to open with the full worked solution, so you can judge the quality first. Focus on the recurring patterns rather than memorizing answers — the same handful of ideas generate most variants.
Are these real quant interview questions?
They are a curated set drawn from our problem bank — the kind of random variables question that actually appears in quant interviews, rewritten for clarity with solutions we author ourselves. We don't claim any single wording is verbatim, and every problem carries a full solution.