Solutions and Intuition
Complete solutions to all practice problems with conceptual explanations.
This section provides complete solutions to all practice problems from Module 9. Each solution includes not just the answer, but the reasoning and intuition behind it. Click on each problem to reveal its solution.
How to Use These Solutions
Try each problem yourself before looking at the solution. The struggle is where learning happens. Once you've attempted a problem, compare your approach to the solution and note any gaps in your understanding.
Problem Set 1: Distributions
Foundational problems on computing probabilities and understanding moment existence.
Problem Set 2: Fat Tail Behavior
Understanding the practical consequences of fat tails through simulation and analysis.
Problem Set 3: Estimation
Applying estimation techniques specific to fat-tailed distributions.
Problem Set 4: Conceptual
Applying fat tail thinking to real-world scenarios.
Key Takeaways from the Solutions
- Power laws dominate Gaussians in the tails: Events that are impossible under Gaussian assumptions may be merely uncommon under power laws.
- Moment existence determines everything: Whether mean and variance exist dictates which statistical tools apply.
- Sample statistics are unstable: Under infinite variance, no sample size is "large enough" for reliable estimation.
- Historical data can be deeply misleading: In fat-tailed domains, the worst event in your data set may be a poor guide to future extremes.
- Standard risk measures fail: VaR, Sharpe ratios, and confidence intervals all assume properties that fat tails violate.
The Meta-Lesson
These problems aren't just mathematical exercises. They represent real-world situations where misapplying thin-tailed thinking has led to financial crises, insurance failures, and catastrophic underestimation of risk. Understanding fat tails is not optional for anyone working with data from the real world.