A Simpler Probabilistic Proof of a Wallis-type Formula for the Gamma Function

We simplify our probabilistic proof of a Wallis-type formula for the Gamma function by using Gamma distributions instead of normal distributions.

An Intuitive Proof of the Hahn-Kolmogorov Theorem

The Hahn-Kolmogorov theorem (which is sometimes called the Carathéodory extension theorem) is used to construct various kind of measures. We provide an intuitive proof of this theorem which avoids using Carathéodory’s (rather) unintuitive criterion for measurability.

Perturbing All Points in R^d into a General Position

We prove that there is a way to move each point in less than a prescribed amount, so that no two points are moved to the same point, and the resulting set of points are in a general position (i.e. no three points are in a line, no four points are in a plane, andContinue reading “Perturbing All Points in R^d into a General Position”

Eight Ways to Derive the Characteristic Function of the Normal Distribution

We present eight proofs of where is a standard normal random variable. This fact is more or less equivalent to saying that the Gaussian is its own Fourier transform. We start with proofs that are rather elementary, and move on to more sophisticated proofs.

Borel’s Proof of the Heine-Borel Theorem

We present a polished version of Borel’s proof based on Cantor’s theory of sets. More specifically, we use transfinite recursion and the fact that any strictly decreasing sequence in a well-ordered set is finite in length.