Quantitative research interviews at top financial firms like Citadel and Five Rings are notoriously rigorous, testing not only your mathematical and statistical knowledge, but also your problem-solving acumen and coding skills. In this comprehensive guide, we’ll walk through several real quant research interview questions asked at Citadel, Five Rings, and similar firms. We’ll break down each question, explain the underlying concepts, and provide detailed solutions. Whether you’re preparing for your next quant interview or aiming to deepen your understanding of quantitative research, this article will give you the edge you need.

Quant Research Interview Questions: Citadel and Five Rings

1. What is the Difference Between Gaussian Naive Bayes (GNB) and Logistic Regression? Which Should You Choose?

Understanding Gaussian Naive Bayes (GNB)

Gaussian Naive Bayes is a probabilistic classifier based on Bayes' theorem, assuming independence between features. For continuous variables, GNB assumes that the features follow a Gaussian (normal) distribution. This makes it a fast and often surprisingly robust classifier.

• Assumptions: Feature independence, normal distribution.
• Formula: For a binary classification, the probability that an input $x$ belongs to class $C_k$ is:
  $$ P(C_k|x) = \frac{P(x|C_k)P(C_k)}{P(x)} $$ For each feature $x_i$: $$ P(x_i|C_k) = \frac{1}{\sqrt{2\pi \sigma_{k,i}^2}} \exp\left( -\frac{(x_i - \mu_{k,i})^2}{2\sigma_{k,i}^2} \right) $$

Understanding Logistic Regression

Logistic Regression is a discriminative model that directly estimates the probability of a class given the input features. It models the log-odds as a linear function of the input variables and applies the sigmoid function to map this to the (0,1) interval.

• Formula: $$ P(y=1|x) = \frac{1}{1+\exp(-w^Tx)} $$
• Assumptions: No distributional assumption on features, but assumes a linear relationship between the log-odds and the features.

Comparison Table

When to Use Each Model?

• Use GNB:
  • When features are largely independent.
  • Features are approximately Gaussian distributed.
  • Dataset is small and you want a quick, robust baseline.
• Use Logistic Regression:
  • When features are correlated.
  • You want to model a linear decision boundary.
  • More interpretability is required.

Interview Tip:

Always mention assumptions, and be prepared to justify your model choice with respect to data properties.

2. Probability of Drawing Balls Until One Color Remains

Question Statement

You have r red balls and w white balls in a bag. You draw balls one at a time, without replacement, until only balls of a single color remain (i.e., one color is exhausted). What is the probability that you run out of white balls first?

Solution Approach

This is a classic combinatorial probability problem. The process continues until all balls of one color are drawn; we want the probability that all white balls are drawn before the last red ball.

The key insight is that the process ends as soon as the last white ball is drawn (if that happens before the last red ball). This is equivalent to asking: What is the probability that the last ball drawn is white? Actually, the process ends as soon as one color is exhausted, but the last ball drawn (the one that exhausts a color) determines the outcome.

Detailed Solution

Suppose we draw balls one by one until all balls of one color are gone. The probability that the last ball drawn (i.e., the ball that exhausts the first color) is white is:

• We must draw all w white balls and r red balls, but the process stops as soon as all white balls have been drawn.
• This is equivalent to arranging the sequence of draws so that the last ball drawn is a white ball, and the rest (r + w - 1) balls are any combination of the remaining balls.

Alternatively, the more standard combinatorial approach is:

• We want the probability that the last ball drawn is a white ball, i.e., that the last white ball is drawn before the last red ball.
• This can be calculated by considering all possible orders of the balls and counting the fraction in which the last ball of a specific color appears before the last ball of the other color.

Let’s formalize this. Consider all possible orderings of the balls. In each ordering, the last ball of each color occurs at some position. We are interested in the orderings where the last white ball occurs before the last red ball.

It is a classic result that the probability that the last white ball appears before the last red ball is:

$$ P(\text{white balls exhausted first}) = \frac{r}{r+w} $$

Similarly, the probability that red balls are exhausted first is:

$$ P(\text{red balls exhausted first}) = \frac{w}{r+w} $$

Intuitive Explanation

This can be understood as follows: imagine removing balls one by one at random. The color that has fewer balls left is less likely to survive to the end. When there are only two balls left, one red and one white, the next draw determines which color is eliminated. By symmetry and combinatorics, the formula above holds.

Final Answer

The probability that you run out of white balls first is:

$$ P(\text{white balls exhausted first}) = \frac{r}{r+w} $$

3. Generate Uniform Random Integer from 1 to 7 Using a Random Integer Generator for 1 to 5

Problem Statement

You have access to a function rand5() that returns a random integer from 1 to 5 with equal probability. How can you use this function to generate a random integer from 1 to 7, also with equal probability?

Solution Explanation

This is a classic use of rejection sampling. The goal is to generate a uniform distribution over 7 outcomes using a uniform distribution over 5 outcomes.

1. Generate a larger uniform range:
   • Call rand5() twice to get two random digits.
   • This gives you $5 \times 5 = 25$ equally likely possibilities.
   • Map (i, j) to a single integer between 1 and 25: $$ \text{num} = 5 \times (i-1) + j $$ where $i, j$ are results of rand5().
2. Use rejection sampling:
   • Since 25 is not a multiple of 7, use only the largest multiple of 7 less than 25, which is 21.
   • If the generated number is between 1 and 21, map it to 1–7 using modulo: $$ \text{result} = ((\text{num} - 1) \bmod 7) + 1 $$
   • If the number is between 22 and 25, reject and repeat.

Python Code Example

import random

def rand5():
    return random.randint(1, 5)

def rand7():
    while True:
        num = 5 * (rand5() - 1) + rand5()  # num in 1..25
        if num <= 21:
            return (num - 1) % 7 + 1

Why Does This Work?

Because each number from 1 to 21 is equally likely, and 21 is divisible by 7, mapping them to 1–7 gives a truly uniform distribution.

4. Variance of x When Randomly Generating Points on a Circle or Sphere

Problem 4a: Variance of x on a Circle

Suppose you randomly generate points $(x, y)$ on the circumference of a unit circle (radius 1, centered at the origin). What is $Var(x)$?

Mathematical Approach

• A random point can be parametrized as: $$ x = \cos \theta, \quad y = \sin \theta $$ where $\theta$ is uniformly distributed on $[0, 2\pi)$.

Compute $Var(x)$:

• $E[x] = E[\cos \theta] = 0$ (since $\cos \theta$ is symmetric around the circle)
• $E[x^2] = E[\cos^2 \theta] = \frac{1}{2\pi} \int_0^{2\pi} \cos^2 \theta \, d\theta$
  • Recall $\cos^2 \theta = \frac{1+\cos 2\theta}{2}$, so: $$ E[x^2] = \frac{1}{2\pi} \int_0^{2\pi} \frac{1 + \cos 2\theta}{2} d\theta = \frac{1}{2} $$
• Therefore, $$ Var(x) = E[x^2] - (E[x])^2 = \frac{1}{2} $$

Final Answer for the Circle

$$ Var(x) = \frac{1}{2} $$ for points uniformly distributed on the circumference of a unit circle.

Problem 4b: Variance of x on a Sphere

Now, suppose you randomly generate points on the surface of a unit sphere (centered at the origin in 3D). What is $Var(x)$?

Mathematical Approach

• A random point can be parametrized as:
  • $x = \sin \phi \cos \theta$
  • $y = \sin \phi \sin \theta$
  • $z = \cos \phi$
  • $\theta \in [0, 2\pi)$, $\phi \in [0, \pi]$
• Uniform distribution on the sphere requires $\phi$ to be distributed such that $\cos \phi$ is uniform on $[-1,1]$ (or, equivalently, $\phi$ has the PDF $p(\phi) = \frac{1}{2} \sin \phi$).

Compute $E[x]$:

• $E[x] = 0$ (by symmetry).
• $E[x^2] = E[\sin^2 \phi \cos^2 \theta] = E[\sin^2 \phi] \cdot E[\cos^2 \theta]$
  • $E[\cos^2 \theta] = \frac{1}{2}$
  • $E[\sin^2 \phi]$ with $p(\phi) = \frac{1}{2} \sin \phi$: $$ E[\sin^2 \phi] = \int_{0}^{\pi} \sin^2 \phi \cdot \frac{1}{2} \sin \phi \, d\phi = \frac{1}{2} \int_0^{\pi} \sin^3 \phi \, d\phi $$
  • $\int_0^{\pi} \sin^3 \phi \, d\phi = \frac{4}{3}$
  • So, $$ E[\sin^2 \phi] = \frac{1}{2} \cdot \frac{4}{3} = \frac{2}{3} $$
  • Thus, $$ E[x^2] = \frac{2}{3} \cdot \frac{1}{2} = \frac{1}{3} $$
• Therefore, $$ Var(x) = E[x^2] - (E[x])^2 = \frac{1}{3} $$

Final Answer for the Sphere

$$ Var(x) = \frac{1}{3} $$ for points uniformly distributed on the surface of a unit sphere.

Summary Table

Conceptual Insights

• Symmetry: The mean of $x$ is zero in both cases due to symmetry.
• Variance decreases as dimension increases: For higher-dimensional spheres, the variance of each coordinate continues to decrease.
• For an $n$-dimensional unit sphere, $Var(x) = \frac{1}{n}$ for each coordinate $x_i$.

Conclusion: Mastering Quant Research Interviews at Citadel and Five Rings

Quant research interview questions at elite firms like Citadel and Five Rings are designed to probe your understanding of probability, statistics, machine learning, and strategic thinking. The problems discussed in this guide showcase the variety of challenges you may encounter—from theoretical differences between classification algorithms (GNB vs. logistic regression), to combinatorial probability, randomized algorithms, geometric statistics, and arbitrage strategies.

To excel at these interviews:

• Understand foundational concepts deeply—don’t just memorize formulas.
• Practice articulating your reasoning clearly and concisely.
• Be ready to code simple algorithms on the spot.
• Show your ability to generalize and adapt solutions.

By carefully studying and practicing real quant research interview questions, you'll not only boost your confidence but also develop the analytical mindset necessary for success at top trading and hedge fund firms.

Good luck with your quant interviews—and remember, every challenge is an opportunity to sharpen your edge!

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