Distributions on Chen Kai Blog

Probability and Statistics (2): Random Variables and the Distributions That Matter

Tue, 20 Aug 2024 09:00:00 +0000

After building the axiomatic foundation in the previous article, you might feel like we spent a lot of time talking about sets and subsets. That’s because we did. The machinery of events and sigma-algebras is necessary but austere — it doesn’t give us a natural way to compute averages, measure spread, or fit models to data.

The bridge between abstract probability and applied statistics is the random variable. Once we assign numerical values to outcomes, the entire toolkit of calculus — derivatives, integrals, series — becomes available. And with calculus comes the ability to characterize randomness through a small set of named distributions, each encoding specific assumptions about how the world generates data.

Functional Analysis (11): Distributions and Sobolev Spaces — Generalized Solutions

Thu, 21 Oct 2021 09:00:00 +0000

I want to start with a confession. For years I treated the Dirac delta the way an undergraduate physicist does: as a function that is zero everywhere except at the origin, where it is infinite, and whose integral is one. That description is, of course, mathematical nonsense. No measurable function can have those properties. Yet every quantum mechanics textbook uses $\delta$ on page one, every signal processing course writes $\delta(t)$ for an impulse, and every PDE book invokes Green’s functions $$E$$ satisfying $\Delta E = \delta$ . Either an entire scientific community has been making a fundamental error for a century, or there is a way to make this object rigorous. The latter, obviously — and the way is the theory of distributions.