Probability Fact

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

How to solve for parameters of beta distribution with given mean and variance johndcook.com/blog/2021/04/0…

The central limit theorem didn't get its name until 1920 even though the first version of the theorem was published in 1733.

Correlation (X, Y) = Covariance(X, Y) / (sigma_X sigma_Y) where sigma_X is the standard deviation of X, sigma_Y = s.d. of Y.

Probability is subtle johndcook.com/blog/2014/01/2…

“A thing appears random only through the incompleteness of our knowledge.” -- Spinoza

A normal random variable is within two standard deviations of its mean about 95% of the time.

If X ~ gamma(n, b) and n is an integer, then P(X <= x) = P(Y >= n) where Y ~ Poisson(x/b).

Suppose g is increasing, X is a r.v. and Y = g(X). Then the PDF f_Y(y) equals f_X( g^-1(y) ) (d/dy) g^-1(y).

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

The exponential distribution is memoryless. If X ~ exponential, P(X > s | X > t) = P(X > s-t).

A Short Introduction to Optimal Transport and Wasserstein Distance alexhwilliams.info/itsneuronalblo…

Covariance(X, Y) = E( (X - μ_X)(Y - μ_Y) ) where μ_X and μ_Y are the means of X and Y.

Richard Stanley's 'Twelvefold Way' for combinatorics johndcook.com//TwelvefoldWay… [pdf]

A binomial(n, p) random variable has mean np and variance np(1-p).

X ~ Fréchet => 1/X ~ Weibull. For this reason the Fréchet distribution is also called the inverse Weibull distribution.

The variance of a Beta(a, b) random variable is ab/((a+b)²(a+b+1)). If a and b are large, this is approximately ab/(a+b)³.

The mean of a Beta(a, b) random variable is a/(a+b). The mode is (a-1)/(a+b-2) provided a > 1 and b > 1.

Stirling's approximation en.wikipedia.org/wiki/Stirling%…

Approximating a logistic distribution with a normal, and vice versa. johndcook.com/blog/2023/11/0…

If X takes only non-negative integer values, the probability generating function of X is G_X(z) = E(z^X).