Quantitative researcher interviews at firms like Susquehanna International Group (SIG) are known for their rigorous mathematical and probability-based questions. One classic problem that tests both your mathematical intuition and technical skills is computing the expected value of a function of a random variable, particularly when the variable is uniformly distributed. In this article, we’ll delve deeply into the question: “Let \( X \sim \text{Uniform}(0,1) \). Compute \( \mathbb{E}[X^3] \).” We will not only solve the problem step by step, but also explore the underlying concepts, variations, and practical implications relevant to quant research interviews.

SIG Quantitative Researcher Interview Question: Expected Value of a Uniform Random Variable

Table of Contents

• Understanding the Uniform Distribution
• What is Expected Value?
• The Interview Problem Statement
• Step-by-Step Solution
• Detailed Mathjax Explanation
• Intuition and Visualization
• Generalizing to \( \mathbb{E}[X^n] \)
• Python Simulation for Verification
• Tips for SIG Quantitative Researcher Interviews
• Conclusion

Understanding the Uniform Distribution

Before solving the problem, it’s essential to understand the core concepts behind the uniform distribution, which is one of the most fundamental probability distributions in quantitative finance, statistics, and probability theory.

Definition

A random variable \( X \) is said to follow a continuous uniform distribution on the interval \([a, b]\), written \( X \sim \text{Uniform}(a, b) \), if it has a constant probability density function (PDF) over that interval.

For our problem, we focus on \( X \sim \text{Uniform}(0,1) \), which is the standard uniform distribution.

Probability Density Function (PDF)

The PDF for \( X \sim \text{Uniform}(0,1) \) is:

\[ f_X(x) = \begin{cases} 1 & \text{if } 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases} \]

What is Expected Value?

The expected value (sometimes called the mean) of a random variable is a fundamental concept in probability theory and statistics, representing the average or “long-term” value of the variable over many trials.

Formal Definition

For a continuous random variable \( X \) with PDF \( f_X(x) \), the expected value of a function \( g(X) \) is:

\[ \mathbb{E}[g(X)] = \int_{-\infty}^{\infty} g(x) f_X(x) dx \]

In our problem, we are asked to compute \( \mathbb{E}[X^3] \), i.e., \( g(x) = x^3 \).

The Interview Problem Statement

Question: Let \( X \sim \text{Uniform}(0, 1) \). Compute \( \mathbb{E}[X^3] \).

Key Points

• This is a classic expectation calculation for a function of a random variable.
• Requires knowledge of integration, PDFs, and properties of the uniform distribution.
• Tests not just calculation skills but the ability to generalize and explain results.

Step-by-Step Solution

Step 1: Write the Expected Value Integral

We know that for \( X \sim \text{Uniform}(0,1) \), the PDF is \( f_X(x) = 1 \) for \( x \in [0, 1] \). Thus,

\[ \mathbb{E}[X^3] = \int_{0}^{1} x^3 \cdot 1 \, dx \]

Step 2: Solve the Integral

The integral of \( x^3 \) from 0 to 1 is a basic calculus problem:

\[ \int x^3 dx = \frac{x^4}{4} + C \]

Applying the limits from 0 to 1:

\[ \int_{0}^{1} x^3 dx = \left[ \frac{x^4}{4} \right]_{0}^{1} = \frac{1^4}{4} - \frac{0^4}{4} = \frac{1}{4} \]

Step 3: State the Final Answer

\[ \boxed{\mathbb{E}[X^3] = \frac{1}{4}} \]

Detailed Mathjax Explanation

Let’s break down each step in Mathjax for clarity.

1. Setting Up the Expectation

\[ \mathbb{E}[X^3] = \int_{-\infty}^{\infty} x^3 f_X(x) dx \] But since \( f_X(x) = 1 \) for \( x \in [0,1] \) and 0 otherwise, this reduces to: \[ \mathbb{E}[X^3] = \int_{0}^{1} x^3 dx \]

2. Evaluating the Integral

Recall the power rule for integration: \[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \] Here, \( n = 3 \): \[ \int x^3 dx = \frac{x^4}{4} + C \] So, \[ \int_{0}^{1} x^3 dx = \left[ \frac{x^4}{4} \right]_0^1 = \frac{1}{4} - 0 = \frac{1}{4} \]

3. Final Answer

\[ \boxed{\mathbb{E}[X^3] = \frac{1}{4}} \]

Intuition and Visualization

Why is the Expected Value \( \frac{1}{4} \)?

For the uniform distribution on \([0,1]\), the mean \( \mathbb{E}[X] \) is \( 0.5 \). However, \( X^3 \) is a convex function, so its average value skews lower, since values less than 1, when cubed, yield even smaller numbers. This “pulls down” the expected value.

Graphical Representation

Let’s visualize both \( X \) and \( X^3 \) for \( X \in [0,1] \).

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(0,1,100)
plt.plot(x, x, label='X')
plt.plot(x, x**3, label='X^3')
plt.legend()
plt.title("X vs X^3 on [0,1]")
plt.xlabel("X")
plt.ylabel("Value")
plt.grid(True)
plt.show()

This plot shows that \( X^3 \) rises much more slowly than \( X \), explaining why \( \mathbb{E}[X^3] \) is only \( \frac{1}{4} \) compared to \( \mathbb{E}[X] = 0.5 \).

Generalizing to \( \mathbb{E}[X^n] \)

Let’s generalize. What if we want \( \mathbb{E}[X^n] \) for \( X \sim \text{Uniform}(0,1) \)?

\[ \mathbb{E}[X^n] = \int_{0}^{1} x^n dx = \left[ \frac{x^{n+1}}{n+1} \right]_0^1 = \frac{1}{n+1} \]

This result is widely applicable in probability and statistics, especially in moments calculation.

Generalization to \( X \sim \text{Uniform}(a, b) \)

If \( X \sim \text{Uniform}(a, b) \), then:

\[ \mathbb{E}[X^n] = \int_{a}^{b} x^n \frac{1}{b-a} dx = \frac{1}{b-a} \left[ \frac{x^{n+1}}{n+1} \right]_a^b = \frac{b^{n+1} - a^{n+1}}{(n+1)(b-a)} \]

Python Simulation for Verification

Interviewers may ask how you’d check your answer numerically. A quick Monte Carlo simulation can verify our analytical result.

import numpy as np

# Simulate 1 million samples from Uniform(0,1)
samples = np.random.uniform(0, 1, 1000000)
expected_value = np.mean(samples ** 3)
print(f"Simulated E[X^3]: {expected_value:.6f}")

Expected output will be close to 0.25, confirming our analytical result.

Tips for SIG Quantitative Researcher Interviews

1. Understand the Basics

• Review fundamental distributions: uniform, normal, exponential, etc.
• Be comfortable with expectation, variance, and higher order moments.

2. Show Generalization

• After solving a specific case (\( n=3 \)), immediately generalize to \( n \) and, if possible, to the case \( X \sim \text{Uniform}(a, b) \).

3. Explain Your Reasoning Clearly

• Walk through the setup, integration, and interpretation step by step.
• Use proper notation and terminology.

4. Validate with a Simulation

• Show that you can quickly check answers using code (Python, R, etc.).

5. Relate Back to Quantitative Finance

• Discuss how such expectations arise in Monte Carlo simulations, option pricing, and risk management.

Conclusion

Computing the expected value of a function of a random variable, such as \( \mathbb{E}[X^3] \) for \( X \sim \text{Uniform}(0,1) \), is a staple of quantitative finance interviews at firms like SIG. This problem tests not only your calculus skills but your understanding of probability distributions, generalization, and practical verification methods. The answer:

\[ \boxed{\mathbb{E}[X^3] = \frac{1}{4}} \]

By mastering such problems, not only will you perform strongly in interviews, but you’ll also build a solid foundation for more advanced concepts in quantitative research and modeling.

Further Reading:

• Expected Value (Wikipedia)
• Uniform Distribution (Wikipedia)
• SIG Interview Preparation