In this article, we will solve and deeply explain several real quant interview questions from SIG and other firms. Through detailed solutions, you will learn the fundamental concepts and problem-solving strategies that top firms expect from successful quant candidates.

Quant Interview Questions from SIG and Other Companies – Solutions & Explanations

1. Expected Value of Primes in 6 Draws from a Custom Deck (SIG)

Problem Statement

A deck of 18 custom playing cards is numbered from 1 to 18. A card is drawn 6 times with replacement. If \( X \) is the number of draws that result in a prime number, what is \( \mathbb{E}(X) \) (the expected value of \( X \))?

Step-by-Step Solution

Step 1: Identify the Prime Numbers Between 1 and 18

First, list all prime numbers in the range 1 to 18:

primes = []
for n in range(1, 19):
    if n > 1:
        for i in range(2, n):
            if (n % i) == 0:
                break
        else:
            primes.append(n)
print(primes)

Calculating:

• 2
• 3
• 5
• 7
• 11
• 13
• 17

So, the primes are: 2, 3, 5, 7, 11, 13, 17 (7 primes).

Step 2: Probability of Drawing a Prime Number

There are 18 cards. The probability that a single draw yields a prime:

\[ P(\text{prime}) = \frac{7}{18} \]

Step 3: Define the Random Variables

Let \( X_i \) be the indicator variable for the \( i \)-th draw: \[ X_i = \begin{cases} 1 & \text{if the } i\text{th draw is a prime} \\ 0 & \text{otherwise} \end{cases} \]

Total number of primes drawn in 6 draws: \[ X = \sum_{i=1}^6 X_i \]

Step 4: Compute the Expected Value

By the linearity of expectation: \[ \mathbb{E}(X) = \sum_{i=1}^6 \mathbb{E}(X_i) \] Each \( X_i \) is Bernoulli with probability \( p = \frac{7}{18} \), so \( \mathbb{E}(X_i) = p \).

Therefore, \[ \mathbb{E}(X) = 6 \times \frac{7}{18} = \frac{42}{18} = \frac{7}{3} \approx 2.333 \]

Step 5: Key Concepts Explained

• Linearity of Expectation: The expected value of a sum is the sum of expected values, regardless of independence.
• Indicator Variables: Useful in counting expected occurrences of an event.
• Bernoulli Trials: Each draw is independent and has two outcomes (prime or not).

Final Answer

\[ \boxed{\frac{7}{3}} \] is the expected number of times a prime number is drawn in 6 draws with replacement from a deck numbered 1 to 18.

2. Expected Distance Between Two Uniform Points on a Unit Circle (SIG)

Problem Statement

Two points are uniformly distributed around the circumference of a unit circle. What is the expected value of the distance between them?

Step-by-Step Solution

Step 1: Parameterize Points on the Circle

Let the two points have angles \( \theta_1 \) and \( \theta_2 \), both uniformly distributed in [0, \( 2\pi \)].

Let’s denote their coordinates:

• Point 1: \( (x_1, y_1) = (\cos \theta_1, \sin \theta_1) \)
• Point 2: \( (x_2, y_2) = (\cos \theta_2, \sin \theta_2) \)

Step 2: Distance Formula

The Euclidean distance between the two points: \[ D = \sqrt{(\cos\theta_1 - \cos\theta_2)^2 + (\sin\theta_1 - \sin\theta_2)^2} \] Expand: \[ D^2 = (\cos\theta_1 - \cos\theta_2)^2 + (\sin\theta_1 - \sin\theta_2)^2 \] \[ = (\cos^2\theta_1 - 2\cos\theta_1\cos\theta_2 + \cos^2\theta_2) + (\sin^2\theta_1 - 2\sin\theta_1\sin\theta_2 + \sin^2\theta_2) \] \[ = (\cos^2\theta_1 + \sin^2\theta_1) + (\cos^2\theta_2 + \sin^2\theta_2) - 2(\cos\theta_1\cos\theta_2 + \sin\theta_1\sin\theta_2) \] \[ = 1 + 1 - 2 [\cos\theta_1\cos\theta_2 + \sin\theta_1\sin\theta_2] \] Recall, \[ \cos\theta_1\cos\theta_2 + \sin\theta_1\sin\theta_2 = \cos(\theta_1 - \theta_2) \] So: \[ D^2 = 2(1 - \cos(\theta_1 - \theta_2)) \] \[ D = \sqrt{2(1 - \cos(\theta_1 - \theta_2))} \]

Step 3: Distribution of Angle Difference

Let \( \Delta = \theta_1 - \theta_2 \). Since both angles are independent and uniform, \( \Delta \) is also uniform on [0, \( 2\pi \)].

So, the problem reduces to finding: \[ \mathbb{E}[D] = \frac{1}{2\pi} \int_0^{2\pi} \sqrt{2(1 - \cos \Delta)} d\Delta \]

Step 4: Simplify the Expression

Recall the trigonometric identity: \[ 1 - \cos\Delta = 2\sin^2 \left( \frac{\Delta}{2} \right) \] So: \[ \sqrt{2(1 - \cos\Delta)} = \sqrt{2 \times 2\sin^2\left( \frac{\Delta}{2} \right)} = 2|\sin(\Delta/2)| \] Since \( \sin(\Delta/2) \) is non-negative for \( \Delta \in [0, \pi] \), and negative for \( \Delta \in [\pi, 2\pi] \), but the absolute value is symmetric.

Step 5: Compute the Integral

\[ \mathbb{E}[D] = \frac{1}{2\pi} \int_0^{2\pi} 2|\sin(\Delta/2)| d\Delta \] \[ = \frac{1}{\pi} \int_0^{2\pi} |\sin(\Delta/2)| d\Delta \]

Now, since \( |\sin(\Delta/2)| \) has period \( 2\pi \), and is symmetric about \( \pi \): \[ \int_0^{2\pi} |\sin(\Delta/2)| d\Delta = 4 \] To show this, substitute \( u = \Delta/2 \), \( d\Delta = 2 du \): \[ \int_0^{2\pi} |\sin(\Delta/2)| d\Delta = 2 \int_0^{\pi} |\sin u| du = 2 \times 2 = 4 \] Because \[ \int_0^{\pi} |\sin u| du = \int_0^{\pi} \sin u du = [-\cos u]_0^{\pi} = -\cos \pi + \cos 0 = 1 + 1 = 2 \]

Step 6: Final Calculation

Therefore, \[ \mathbb{E}[D] = \frac{1}{\pi} \times 4 = \frac{4}{\pi} \approx 1.273 \]

Key Concepts Used:

• Parameterization of points on a circle
• Distance formula in 2D space
• Expected value as an integral over uniform distribution
• Trigonometric identities and substitutions

Final Answer

\[ \boxed{\frac{4}{\pi}} \] is the expected value of the distance between two points chosen uniformly at random on the circumference of the unit circle.

3. Effect of Doubling Your Sample Size in Regression (Sunrise Futures)

Problem Statement

What happens if you double your sample when you do regression? What happens to the coefficients? What happens to \( R^2 \) and p-values?

Step-by-Step Conceptual Explanation

Step 1: The Regression Model Setup

Consider a standard linear regression model: \[ y = X\beta + \epsilon \] where:

• \( y \) is an \( n \times 1 \) vector of responses
• \( X \) is an \( n \times p \) matrix of predictors
• \( \beta \) is a \( p \times 1 \) vector of coefficients
• \( \epsilon \) is an \( n \times 1 \) vector of errors (\( \epsilon \sim N(0, \sigma^2 I) \))

Step 2: Effect on Coefficient Estimates

The OLS estimator is: \[ \hat{\beta} = (X^TX)^{-1} X^T y \] Doubling sample size means you have twice as many data points, say from \( n \) to \( 2n \).

• If new data are independent and identically distributed (i.i.d.), the expected value of \( \hat{\beta} \) does not change in expectation. The coefficients may change slightly due to sampling variation, but they are still unbiased estimators of the true \( \beta \).
• Variance of the Coefficient Estimates: The variance decreases as sample size increases. Specifically: \[ \text{Var}(\hat{\beta}) = \sigma^2 (X^T X)^{-1} \] When you double the number of i.i.d. samples, the elements of \( X^T X \) double, making the variance of \( \hat{\beta} \) roughly halve.

Step 3: Effect on \( R^2 \) (Coefficient of Determination)

\[ R^2 = 1 - \frac{\text{SSR}}{\text{SST}} \] SSR: sum of squared residuals; SST: total sum of squares.

• If the new data are drawn from the same distribution, the population \( R^2 \) does not change. The sample \( R^2 \) may become more stable and less sensitive to outliers.
• If the new data differ in distribution, \( R^2 \) could increase or decrease depending on how well the model fits the expanded data.

Step 4: Effect on p-values

P-values for coefficients are computed as: \[ p = 2 \left[1 - F\left(\left|\frac{\hat{\beta}_j}{\text{SE}(\hat{\beta}_j)}\right|\right)\right] \] where \( SE(\hat{\beta}_j) \) is the standard error of the coefficient.

• Standard errors decrease as sample size increases (typically scale with \( 1/\sqrt{n} \)), so the t-statistics increase.
• P-values decrease for nonzero coefficients, making it easier to reject the null hypothesis (i.e., more statistically significant results).

Step 5: Summarizing the Effects

• Coefficients (\( \hat{\beta} \)): No systematic change in expectation if new data are i.i.d., but estimates become more stable (lower variance).
• Variance of Coefficients: Approximately halves, making confidence intervals narrower.
• \( R^2 \): No expected change if data are from the same distribution, but the estimate becomes more robust (less sensitive to noise or outliers).
• p-values: Decrease for nonzero coefficients, leading to higher statistical power (easier to detect true effects).

Step 6: Mathematical Illustration

Assume simple linear regression with \( n \) observations. The standard error of the coefficient \( \hat{\beta}_1 \) is: \[ SE(\hat{\beta}_1) = \sqrt{\frac{\sigma^2}{\sum (x_i - \bar{x})^2}} \] If you double \( n \) (i.e., collect twice as many i.i.d. samples), the denominator approximately doubles, so \( SE(\hat{\beta}_1) \) decreases by about \( 1/\sqrt{2} \).

Thus, the t-statistic for \( \hat{\beta}_1 \) increases, and the associated p-value decreases, assuming the effect size stays the same. This is the essence of increased statistical power with more data.

Step 7: Simulation Example

import numpy as np
import statsmodels.api as sm

# Generate data
np.random.seed(0)
n = 100
X = np.random.randn(n, 1)
beta = 2.0
y = X.flatten()*beta + np.random.randn(n)

# Fit regression
X_const = sm.add_constant(X)
model = sm.OLS(y, X_const).fit()
print("Original sample size:", n)
print(model.summary())

# Double the sample
X2 = np.random.randn(n, 1)
y2 = X2.flatten()*beta + np.random.randn(n)
X_all = np.vstack([X, X2])
y_all = np.concatenate([y, y2])
X_all_const = sm.add_constant(X_all)
model2 = sm.OLS(y_all, X_all_const).fit()
print("Doubled sample size:", 2*n)
print(model2.summary())

Running this simulation would show that the coefficient estimates stay close to the true value, but the standard errors and p-values improve with the larger sample size.

4. Key Quant Concepts Illustrated by These Questions

Each of these interview problems highlights a core quantitative concept:

• Linearity of Expectation and Indicator Variables: The prime card problem demonstrates the power of expectation properties and discrete probability.
• Geometric Probability and Integration: The unit circle problem showcases parameterization, trigonometric identities, and the use of integrals in expected value calculations.
• Statistical Inference and Regression: The regression sample size question bridges theory and application, focusing on the impact of data on the reliability and significance of model estimates.

5. Practice and Preparation Tips for Quant Interviews

To excel in quant interviews at SIG and similar firms, keep these strategies in mind:

• Brush up on probability fundamentals—especially expectation, variance, and combinatorics.
• Practice geometric and integration problems—many trading questions require spatial reasoning or continuous mathematics.
• Understand statistical inference deeply—know how OLS works, what affects variance, and how hypothesis testing operates.
• Communicate your reasoning clearly—walk through your logic step by step, as in the explanations above.

6. Additional Sample Quant Interview Questions

Here are a few more examples typical of quant firm interviews:

• Probability: What is the expected number of coin tosses until two consecutive heads appear?
• Estimation: Estimate the probability that a random 2-digit number is divisible by 7.
• Logic: You have two ropes and a lighter. Each rope takes exactly 60 minutes to burn, but not at a uniform rate. How do you measure 45 minutes?
• Statistics: What happens to the confidence interval of a mean if you quadruple your sample size?

7. Conclusion

Quantitative interviews at SIG and peer firms are built around challenging, conceptually rich problems. Mastering these types of questions requires a blend of mathematical knowledge, logical reasoning, and clear communication. By practicing with problems like those solved above—on probability, geometry, and statistics—you can develop the deep understanding and fast analytical thinking that top trading firms are seeking.

Whether you are preparing for a quant interview or just want to sharpen your problem-solving skills, tackling these problems step by step and understanding the underlying concepts is the best way forward. Good luck!

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