Random Variables Interview Questions
Random variables questions in quant interviews come in four recurring flavors: pushing a distribution through a transformation, deciding what a joint distribution does and does not factor into, running covariance–correlation algebra, and reading moments off the named distributions — with order statistics as the harder tail.
The common thread: identify the distribution, push it through the map, and let the moments do the rest.
Last updated 2 July 2026 · sub-areas, difficulty mix and firm attributions on this page are compiled directly from the 49 random-variables problems in the QuantVault bank; firm tags are candidate-reported, not employer-verified, and no individual author is named in our source data.
- Core sub-areas
- Transformations & change of variables · joint distributions & independence · covariance & correlation algebra · named distributions & order statistics
- Typical difficulty
- Centered on medium — 18 easy, 25 medium and 6 hard across the 49-problem set
- Who leans on it
- Research-heavy interview loops — 48 of the 49 problems carry a quant-research role tag, with candidate tags from Citadel, Two Sigma, Five Rings, Renaissance and Tower Research among others
- Practice pool
- 49 problems in the bank · 4 free to open with the full worked solution
Where random variables show up: mostly in quant-research phone screens and technical onsites, where reshaping a distribution cleanly is the test — the same machinery that turns two plain uniforms into a disk-uniform point prices everything downstream. For how it sits inside specific funnels, see the Citadel interview questions guide and the Two Sigma interview questions guide.
The surfaceThe random variables sub-areas quant interviews test
Five families cover essentially every random-variables question in the bank. Each row pairs a sub-area with the recurring question shape and a representative type — a flavor drawn from real problems, not a leaked wording — so you can see where the difficulty actually lives.
| Sub-area | Recurring shape | Representative type (flavor, not a real question) |
|---|---|---|
| Transformations & change of variables | Chase the event “g(X) ≤ t” back through a CDF you know, then differentiate | Derive the density of a square, an exponential of a normal, or a radius — e.g. build a uniform point in a disk from plain uniforms with no rejection step. |
| Joint distributions & independence | Factor the joint density — or prove you can't — before computing anything | A sum-and-difference or correlated-normal pair: decide whether the components are independent, then exploit (or refuse) the factorization. |
| Covariance & correlation algebra | Expand Cov(·,·) by bilinearity; only then reach for numbers | The correlation of two linear combinations of the same variables — the answer is a sign and a ratio, never an integral. |
| Named distributions & their moments | Recognize normal / uniform / Poisson structure and read moments off the shelf | Match a generating function to its distribution, or produce the mean and variance of a linear combination in one line. |
| Order statistics & extremes | Start from P(max ≤ t) = F(t)n and differentiate | The density of the maximum or minimum of n uniforms, or the expected value of the k-th smallest draw. |
What's confirmed vs. what varies: the sub-areas and the difficulty mix above come straight from the problem set, so they are stable. Which firm asks which flavor is candidate-reported through problem tags — treat the firm attributions as directional, and expect research tracks to push deeper into transformations and order statistics than trading tracks, which tend to reroute this material into expected-value and betting questions.
The patternsSignature random variables question patterns
Three moves generate most correct answers in this topic. Each worked box below is a 60–90 second micro-example in the interview's actual cadence — the reasoning template, derived from free problems in the set rather than any firm's wording.
Change of variables — equate CDFs, don't guess densities
Takeaway: don't reach for a Jacobian first — write the event “g(X) ≤ t” as an event about X, evaluate it with the CDF you already know, then differentiate.
Shape. Using only uniform draws \(U, V\) on \([0,1]\), generate a point uniformly distributed in the unit disk with no rejection step. Why is the radius \(R=\sqrt{U}\) — and not \(R=U\) — the right choice?
1. Say what “uniform” means as a CDF. Uniform in the disk means probability proportional to area, so \(P(R \le r) = \pi r^2 / \pi = r^2\). That quadratic CDF is the whole problem.
2. Match the CDF. If \(R = g(U)\) with \(g\) increasing, then \(P(g(U) \le r) = P(U \le g^{-1}(r)) = g^{-1}(r)\). Setting \(g^{-1}(r)=r^2\) forces \(g(u)=\sqrt{u}\).
3. Finish the construction. Draw the angle \(\Theta = 2\pi V\) independently and output \((R\cos\Theta, R\sin\Theta)\). The move: equate CDFs before touching densities — the naive \(R=U\) crowds points near the center, because area grows with \(r^2\), not \(r\).
Correlation algebra — bilinearity before arithmetic
Takeaway: correlation is invariant under shifting and positive rescaling — a linear transformation can only ever flip its sign.
Shape. \(X\) and \(Y\) have correlation \(\rho\). What is the correlation of \(3-2X\) and \(5Y+1\)?
1. Kill the shifts. Covariance ignores constants: \(\mathrm{Cov}(3-2X,\,5Y+1) = (-2)(5)\,\mathrm{Cov}(X,Y)\).
2. Rescale the denominators. The standard deviations scale to \(2\sigma_X\) and \(5\sigma_Y\), so the correlation is \(-10\,\mathrm{Cov}(X,Y) / (10\,\sigma_X \sigma_Y) = -\rho\) — the magnitudes cancel exactly.
3. Read off the general rule. \(\mathrm{corr}(aX+b,\,cY+d) = \mathrm{sign}(ac)\,\rho\). The move: expand by bilinearity once, and every “correlation after a linear map” question reduces to tracking one sign.
Sum and difference — zero correlation is not independence
Takeaway: for i.i.d. \(X, Y\), the sum and the difference are always uncorrelated — but independent only in the special normal case, and interviewers probe exactly that gap.
Shape. \(X\) and \(Y\) are i.i.d. with variance \(\sigma^2\). What is the correlation of \(S = X+Y\) and \(D = X-Y\)?
1. Expand the covariance. \(\mathrm{Cov}(S,D) = \mathrm{Var}(X) - \mathrm{Var}(Y)\) — the cross terms cancel — and equal variances make this zero.
2. Conclude, carefully. Correlation zero, for any common distribution. No normality was used, only matching variances.
3. Name the trap. Uncorrelated is weaker than independent: for fair \(\pm 1\) coin flips, \(D \ne 0\) forces \(S = 0\), so \(D\) carries real information about \(S\). Joint normality is what upgrades zero correlation to independence. The move: cancel the variances, then say the independence caveat out loud — that sentence is the signal.
Free practiceRandom variables practice questions with solutions
The free pool here is small and we would rather keep it honest than pad it: 4 of the 49 random-variables problems are free — three easy and one medium — and each link opens the full worked solution. The hard tail (order statistics, multi-step transformations) sits in the paid tier; the worked patterns above walk its core reasoning shapes.
- Correlation Under Linear TransformationsCorrelation algebraeasyfree
- Correlation of Sum and DifferenceJoint + independenceeasyfree
- Variance and Expectation InequalityMomentseasyfree
- Uniform Point in the Unit Disk Without RejectionChange of variablesmediumfree
For depth beyond these four, the full Random Variables problem bank holds all 49 problems — filterable by difficulty and sub-area tag, each with a complete worked solution.
The planHow to prepare for random variables questions
Five techniques to make reflexive, in the order they pay off. The first three map directly onto the worked patterns above, so you can drill each one immediately after reading it.
- Make the CDF method your first instinct. For any transformation, write “g(X) ≤ t” as an event about X, evaluate, differentiate. Reach for the Jacobian only when the map is genuinely multivariate — and even then, state the CDF version first.
- Drill covariance bilinearity until it is arithmetic. Expanding Cov of two linear combinations term by term should take seconds; every correlation-after-a-linear-map question is one expansion plus a sign check.
- Rehearse the “uncorrelated is not independent” sentence. Know the sum-and-difference cancellation cold, and know that joint normality is the exception that upgrades it — interviewers listen for the caveat, not the zero.
- Memorize the moment kit for the named distributions. Mean, variance and MGF for uniform, normal, exponential and Poisson — recognizing a product of MGFs as an old friend is what turns a ten-minute problem into a one-liner.
- Own the maximum. P(max ≤ t) = F(t)n unlocks most order-statistics questions in one differentiation. Then run the free practice set above and take the interactive random variables playlist on a clock.
FAQRandom variables interview questions — frequently asked
How important are random variables for quant interviews?
Core material for quant-research interviews: 48 of the 49 problems in this set carry a quant-research role tag, and manipulating distributions is the working language of those rounds. Trading loops draw on it too, usually as the bridge into expected-value and betting questions.
What random variables topics should I focus on?
Transformations via the CDF method, joint distributions with independence checks, and covariance–correlation algebra cover the bulk of the set. Add the moment kit for the normal and uniform distributions, plus basic order statistics, to handle the harder tail.
How hard are random variables interview questions?
Centered on medium: of the 49 problems in this set, 18 are easy, 25 are medium and 6 are hard. The hard tail concentrates in order statistics and multi-step transformations, where finding the setup — not the calculus — is the real test.
Are these real quant interview questions?
They are representative, not verbatim. The problems are curated from our bank to match the random-variables question shapes candidates report from quant interviews, rewritten for clarity with worked solutions we author ourselves — we never claim any wording is a leaked question.