Probability Fact @ProbFact
Daily tweets on probability from @JohnDCook. johndcook.com/blog/applied-p… Joined November 2009-
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The exponential distribution is memoryless. If X ~ exponential, P(X > s | X > t) = P(X > s-t).
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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).
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…
Kolmogorov's zero-one law: en.wikipedia.org/wiki/Kolmogoro…
Random is as random does johndcook.com/blog/2012/04/1…
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