Problem Set 1: Distributions
Problems on Pareto, Gaussian, and other distributions.
This problem set tests your understanding of key probability distributions, particularly the Pareto and Gaussian distributions. These problems will help you develop intuition for how different distributions behave, especially in their tails.
Work through each problem carefully. Solutions are provided in Module 10.
Problem 1: Pareto Tail Probability
Problem Statement
If , compute .
Hints
- Recall the Pareto survival function: for
- Substitute the given values for , , and
Why This Matters
This problem illustrates the slow decay of Pareto tails. Compare your answer to what you would get for a Gaussian distribution — the difference is dramatic!
Problem 2: Moment Existence
Problem Statement
For what values of does have a finite variance?
Hints
- Variance requires the second moment to be finite
- For Pareto, exists only when
- Think about what happens to the integral for different values of
Connection to Fat Tails
This is a fundamental result. Many real-world phenomena (wealth distribution, city sizes, market returns) have between 1 and 3, meaning variance may not exist!
Problem 3: Gaussian vs. Power Law
Problem Statement
If and , compare and .
Hints
- For the standard normal, use the approximation (or look up in a table)
- For the Pareto, use the survival function directly
- Calculate the ratio — how many times more likely is the Pareto event?
The Core Message
This comparison captures the essence of fat tails. Events that are virtually impossible under Gaussian assumptions become merely unlikely under power laws. A 5-sigma event in a Pareto world is far more common than in a Gaussian world.
What You Should Learn
- How to compute tail probabilities for Pareto distributions using the survival function
- The relationship between the tail exponent and moment existence
- The dramatic difference between Gaussian and power law tail behavior
- Why these mathematical differences have profound practical implications