Citadel·Data Scientist·Technical Phone Screen·Senior

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Jul 2026

Summary

Citadel data science interview with some heavy probability/stats theory. The question was essentially a two-part derivation involving change of variables and the logistic transform, which felt more like a math exam than a job interview.

Questions Asked (1)

Given a random variable X with density f_X on (0,1) and the logistic function g(u) = 1/(1 + e^{-u}), derive the general density formula for U = g^{-1}(X) using a change of variables. Then, for X ~ Uniform(0,1), compute the distribution, mean, and variance of U = log(X/(1-X)), showing the Jacobian steps explicitly.

Data ModelingTechnical Trade-offsAlgorithms & Data Structures

Author's notes

This wrecked me a little.

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AI HintsAI Generated

Suggested Approach

Begin by establishing the general change-of-variables formula for a monotone transformation, carefully computing the inverse function g^{-1} and its derivative (the Jacobian). Then specialize to the Uniform(0,1) case, identify the resulting distribution as the standard logistic, and derive its mean and variance from known moment formulas or direct integration.

Pro tip: Explicitly verify that g^{-1} is monotonically increasing and that the support maps correctly from (0,1) to (-∞, +∞) before writing the density — Citadel interviewers will probe whether you check regularity conditions, not just mechanically apply formulas.

1. Identify the Inverse Transform and Its Domain

Note that g(u) = 1/(1+e^{-u}) is the logistic (sigmoid) function mapping ℝ → (0,1), so U = g^{-1}(X) = log(X/(1-X)) is the log-odds (logit) function mapping (0,1) → ℝ. Confirm the transformation is strictly monotone increasing, which guarantees a clean change-of-variables.

2. Derive the General Density via Change of Variables

Apply the formula f_U(u) = f_X(g(u)) · |dg(u)/du|. Compute the Jacobian: dg/du = e^{-u}/(1+e^{-u})^2 = g(u)(1−g(u)), so f_U(u) = f_X(g(u)) · g(u)(1−g(u)) for u ∈ ℝ.

3. Specialize to X ~ Uniform(0,1)

Substitute f_X(x) = 1 on (0,1) into the general formula: f_U(u) = 1 · g(u)(1−g(u)) = e^{-u}/(1+e^{-u})^2, which is exactly the standard logistic distribution density. State explicitly that U ~ Logistic(0,1).

4. Compute Mean and Variance

The standard logistic distribution has mean 0 (by symmetry of f_U around u=0, since f_U(u) = f_U(−u)) and variance π²/3. If required to show this from scratch, use the moment generating function M(t) = πt/sin(πt) or integrate by parts, citing the result E[U²] = π²/3.

5. Sanity-Check and Contextualize

Verify the density integrates to 1 and note the practical relevance: this derivation underpins logistic regression's probabilistic interpretation, the probit-logit connection, and simulation via inverse CDF sampling — all directly relevant to quantitative modeling at a firm like Citadel.

Key Points to Mention

The Jacobian term |dg/du| = g(u)(1−g(u)) = e^{-u}/(1+e^{-u})^2 and how it arises from differentiating the logistic function
Monotonicity of g ensures the change-of-variables formula applies without needing to split the integral or take absolute values beyond the standard form
Recognition that the resulting distribution is the standard Logistic(0,1) with support on all of ℝ
Mean = 0 justified by symmetry (f_U is symmetric about 0), and Variance = π²/3 with at least a sketch of derivation
Connection to inverse CDF (quantile) sampling: since X ~ Uniform(0,1), U = logit(X) is a standard method to simulate logistic random variables
Broader modeling relevance: logit link functions in GLMs, log-odds interpretation in binary classification, and heavy-tailed behavior of the logistic vs. normal distribution

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