Classifying Probability Distribution Functions (UMAP)

In this module, the Cantor set is used to construct a distribution that is not discrete or absolutely continuous. An urn model illustrates a random variable which has the Cantor distribution. This distribution leads naturally to the concept of a singular continuous function. An arbitrary distribution function can always be decomposed uniquely into a convex linear combination of three distributions: discrete, absolutely continuous, and singular continuous. Necessary and sufficient conditions are provided for a distribution function to be absolutely continuous, rather than singular continuous. Some of these results can be generalized to functions other than probability distributions.

Table of Contents:

PROLOGUE

1. INTRODUCTION

2. DISTRIBUTION FUNCTIONS

3. THE CANTOR FUNCTION
3.1 Constructing the Cantor Set
3.2 Defining the Cantor Function
3.3 Properties
3.4 A Probability Interpretation

4. SINGULAR CONTINUOUS DISTRIBUTIONS
4.1 Definition
4.2 A Remarkable Example

5. ABSOLUTELY CONTINUOUS DISTRIBUTION
5.1 A Preliminary Theorem
5.2 Relating to the Fundamental Theorem of Calculus
5.3 A Characterization

6. SOME REAL ANALYSIS: BOUNDED VARIATION

7. THE CLASSIFICATION THEOREM

8. DISCUSSION

9. REFERENCES

10. ANSWERS TO THE EXERCISES