CauchyDistribution

The Cauchy distribution with location $a$ and scale $b$ — heavy-tailed enough that its mean and variance don't exist.

CauchyDistribution(a, b)an inert distribution object — carries

a, b

, unevaluated.

CauchyDistribution()the standard Cauchy distribution,

a=0, b=1

Typed signatures

Type	Package	Applies
`(real?, real<0..>?) -> distribution`	`statistics`	always

Domain: Statistics

Details

$PDF(x) = 1/(\pi b(1+((x-a)/b)^2))$ , exact.
$CDF(x) = 1/2 + \arctan((x-a)/b)/\pi$ , exact — added in the batch that also filled the matching gap for StudentTDistribution and HypergeometricDistribution (each has its own closed or exact-finite-sum form; none shares a formula with Cauchy).
$Mean$ and $Variance$ are Indeterminate — they genuinely do not exist for this distribution, matching Wolfram.
CharacteristicFunction $(t) = e^{iat-b|t|}$ , exact. No MomentGeneratingFunction — $E[e^{tX}]$ diverges for any $t\neq0$ , a genuine fact about this distribution, not a gap.
RandomVariate samples via inverse transform: $a + b\tan(\pi(U-1/2))$ .