Cantor Singular Distribution
Build a continuous singular distribution from finite ternary bit sums.
Free theory articleBuild a continuous singular distribution from finite ternary bit sums.
Concept & interpretation
The Cantor distribution is a continuous but non-absolutely continuous probability distribution. It acts as a counterexample in measure theory and analysis, demonstrating that a continuous cumulative distribution function (CDF) can have a derivative of exactly zero almost everywhere.
Interactive Experiment
Explore the convergence of the discrete finite approximations to the singular continuous limit using the module. Follow these steps:
- Open the Cantor Singular Distribution module.
- Observe the default state with Depth set to 4 and Probe x at 1/3. Notice how the Limit CDF and Finite CDF align.
- Change Depth to 1 and Probe x to 1/9. Observe that the Finite CDF is 0.5 because 1/9 >= 0, but it is smaller than 2/3, while the Limit CDF equals 0.25.
- Adjust Depth to higher values (e.g., 6 or 8) and observe how the step-like Finite CDF tightly bounds and approaches the smooth fractal limit curve.