Agnesi died in 1799 without imagining that her curve would have a second life, far from geometry. In the nineteenth century it reappeared in probability theory, as one of the most treacherous examples in statistics.

A probability density

We saw in The area under the curve that the total area under $y = \frac{1}{1 + x^2}$ is $\pi$. Dividing by $\pi$, the area becomes 1, and the curve becomes a probability density:

$$f(x) = \frac{1}{\pi(1 + x^2)}.$$

This is the Cauchy distribution. The chance that a draw falls between $c$ and $d$ is the area under the curve over that interval, $\frac{1}{\pi}(\arctan d - \arctan c)$. Drawn, it looks very much like the normal curve: a symmetric bell centred on zero. The difference lies in the tails. The normal falls off like $e^{-x^2/2}$, too fast for enormous values to appear; the curve of Agnesi falls off only like $\frac{1}{x^2}$.

The mean that does not exist

This difference has an astonishing consequence. The mean of a distribution is the integral of $x\,f(x)$. For the Cauchy distribution, the integral over the positive half is

$$\int_0^{X} \frac{x}{\pi(1 + x^2)}\,dx = \frac{1}{2\pi}\ln(1 + X^2),$$

which grows without limit as $X$ grows. The negative part does the same in the other direction, and the calculation becomes “infinity minus infinity”: the distribution has no mean. It has no variance either.

In practice, this means that the average of many draws does not settle down. The law of large numbers and the central limit theorem, which require a finite mean and variance, simply do not apply. Worse: one can prove that the average of $n$ Cauchy draws has exactly the same distribution as a single draw. Gathering more data does not help at all.

The median, on the other hand, works very well: half the area lies to the left of zero and half to the right, and the median of many draws approaches zero. That is why statisticians use the curve of Agnesi as a warning: before computing averages, it is good to know whether the tails of the data are light or heavy.

Poisson, Cauchy and physics

The first to notice the problem was Siméon-Denis Poisson, in 1824, who used this distribution as a counterexample to Laplace’s theory of errors. The name came from Augustin-Louis Cauchy, who discussed it in 1853 in a dispute with Irénée-Jules Bienaymé over the method of least squares.

In physics the same curve appears under another name, the Lorentzian, after Hendrik Lorentz. It describes the shape of the lines in the spectra of atoms and molecules, the response of an oscillator near resonance and, in particle physics, the shape of the peaks that reveal unstable particles (the Breit–Wigner formula). Whenever something oscillates and loses energy little by little, the witch of Agnesi tends to appear.

Exercise: the quartiles

What is the probability that a Cauchy draw falls between $-1$ and $1$?

$\frac{1}{\pi}(\arctan 1 - \arctan(-1)) = \frac{1}{\pi}\cdot\frac{\pi}{2} = \frac{1}{2}$. Half of the draws fall between $-1$ and $1$: these are the quartiles of the distribution. For the standard normal, half fall between $-0.674$ and $0.674$; the Cauchy distribution is more “spread out” in the middle and much more so in the tails.

Exercise: the lighthouse

In the lighthouse problem, the angle $\theta$ between the pulse and the perpendicular to the beach is drawn uniformly between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$, and the pulse hits the beach at the point $x = \tan\theta$. Show that $x$ has density $\frac{1}{\pi(1 + x^2)}$.

The probability that the pulse lands to the left of $x$ is the probability that $\theta$ is less than $\arctan x$: $F(x) = \frac{\arctan x + \frac{\pi}{2}}{\pi} = \frac{1}{2} + \frac{\arctan x}{\pi}$. The density is the derivative: $F'(x) = \frac{1}{\pi}\cdot\frac{1}{1 + x^2}$. It is the curve of Agnesi divided by $\pi$.

Sources: Siméon-Denis Poisson, “Sur la probabilité des résultats moyens des observations”, Connaissance des temps (1824) · Augustin-Louis Cauchy, “Sur les résultats moyens d’observations de même nature”, Comptes rendus de l’Académie des sciences 37 (1853) · Stephen M. Stigler, “Cauchy and the Witch of Agnesi”, Biometrika 61 (1974).

All topics on Maria Gaetana Agnesi · ← The inverse method of tangents · A woman at the university →

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