If Z ~ normal(0, 1) and W ~ chi-squared(n) are independent then Z/√(W/n) ~ t(n). en.wikipedia.org/wiki/Student%2…

How to find a distribution satisfying two quantile conditions johndcook.com/blog/2010/01/3…

St. Petersburg paradox: en.wikipedia.org/wiki/St._Peter…

A gamma distribution with shape a and scale b has mode b(a-1).

Where to wait for an elevator johndcook.com/blog/2010/11/2…

Poisson was the first to describe what we now call the Cauchy distribution.

Why do probabilists take random variables to be Borel (and not Lebesgue) measurable? mathoverflow.net/questions/3160…

Var(X) = E(X²) - E(X)²

Kolmogorov's zero-one law: en.wikipedia.org/wiki/Kolmogoro…

Random is as random does johndcook.com/blog/2012/04/1…

Oldies but goldies: Andrew Berry, The Accuracy of the Gaussian Approximation to the Sum of Independent Variates, 1941. Provides a quantitative estimation of the convergence speed of the central limit theorem. en.wikipedia.org/wiki/Berry%E2%…

The entropy of a random variable is the integral of -f(x) log( f(x) ) where f(x) is the PDF of X.

If X ~ t with v degrees of freedom, then X² ~ F(1, v).

Relating correlation and eccentricity of contours johndcook.com/blog/2023/07/0…

Borel-Cantelli lemma: en.wikipedia.org/wiki/Borel%E2%…

DasGupta's inequality for normal random variables

If X ~ chi-squared(w) and Y ~ chi-squared(v) are independent then (X/w) / (Y/v) ~ F(w, v).

If X ~ chi-squared(w) and Y ~ chi-squared(v) are independent then X + Y ~ chi-squared(w + v).

First names and Bayes theorem johndcook.com/blog/2023/04/0…

The F distribution is applied in analysis of variance (ANOVA).