Cauchy distribution

The Cauchy distribution looks like a Normal from the centre but has radically heavier tails. Those tails cause everything to break: the mean is undefined, the variance is infinite, and the Law of Large Numbers does not apply.

f (x; x_{0}, γ) = \frac{1}{π γ [ 1 + ( \frac{x - x _{0}}{γ} ) ^{2} ]}

PDF comparison

Normal(0,1) Cauchy(0,1)

Running sample mean

Normal — converges to 0

Cauchy — wanders indefinitely

n (samples) 500

Both distributions look similar near the peak, but Cauchy has far heavier tails. Because the Cauchy mean is undefined, the running mean never settles — unlike Normal where it converges by the

n

law.

What to notice

PDFs look similar near the peak. Both distributions are symmetric and bell-shaped. The difference is in the tails: Cauchy’s tails decay as $1/ x^{2}$ , while Normal’s decay exponentially.
Running means diverge. The Normal running mean converges to zero within a few hundred samples. The Cauchy mean wanders indefinitely, with occasional violent jumps driven by extreme outliers.
Resample to see how different Cauchy runs can look. The wandering is not a quirk of one bad draw — it is intrinsic to the distribution.

Why no mean?

The Cauchy mean integral diverges:

E [X] = \int_{- \infty}^{\infty} x \cdot \frac{1}{π ( 1 + x ^{2} )} d x

The positive and negative tails are equally heavy, so the integral does not converge absolutely. The Law of Large Numbers requires a finite mean — without one, sample averages do not stabilise.

The ratio connection

If X ~ Normal(0,1) and Y ~ Normal(0,1) independently, then X/Y ~ Cauchy(0,1). Dividing by a near-zero normal draw can give an arbitrarily large result, which is why heavy tails appear in ratios.