Probability Fact

Putting probability to work johndcook.com/blog/applied-p…

What probabilists call 'characteristic functions' everyone else calls Fourier transforms.

Distributions are uniquely determined by their moment generating functions if the MGF exists in a neighborhood of zero.

Simple normal tail probability estimate johndcook.com/blog/2021/11/0…

Chebyshev published the famous inequality that bears his name in 1866.

Two definitions of expectation johndcook.com/blog/2008/06/2…

If X and Y are independent, the moment generating function (MGF) of X+Y is the product of the MGF's of X and Y.

The central limit theorem began in 1733 when de Moivre approximated binomial probabilities using the integral of exp(-x²).

A Weibull random variable can have increasing, constant, or decreasing hazard.

The Burr distribution johndcook.com/blog/2023/02/1…

The exponential distribution has constant hazard.

New post: Posterior variance johndcook.com/blog/2026/07/1…

If X has mean μ and variance σ², then (X - μ)/σ has mean 0 and variance 1.

The product of two independent log-normal random variables has a log-normal distribution.

Bayes theorem is easy to prove, hard to understand.

Queueing and economies of scale johndcook.com/blog/2022/01/1…

The 80-20 rule corresponds to a Pareto probability distribution with shape log(5)/log(4) = 1.161.

Legendre published the method of least squares in 1805.

Var(X) = Var( E(X|Y) ) + E( Var(X|Y) )

The Pearson probability distributions johndcook.com/blog/2023/02/1…

Var(X + Y) = Var(X) + Var(Y) + 2 Cov(X, Y)