The Laplace Distribution

Take two exponential decay curves — the kind that describes radioactive decay or waiting times — and glue them back to back at a shared center, one decaying to the left and one to the right. That's the Laplace distribution: symmetric like the normal curve, but with a sharp peak (not a smooth rounded top) and tails that decay linearly-in-log rather than quadratically, meaning genuinely extreme values show up far more often than a normal curve would predict.

Its density is $f(x) = \frac{1}{2b}\exp\left(-\frac{|x-\mu|}{b}\right)$, where $\mu$ is the center and $b$ controls the spread — larger $b$ means a wider, flatter distribution. The key contrast with the normal curve: the normal's exponent uses $(x-\mu)^2$, which crushes far-out values almost to nothing, while the Laplace's exponent uses $|x-\mu|$, which crushes them much more gently. That single difference is why the Laplace has noticeably fatter tails.

Worked example. Daily equity returns often cluster tightly near zero with occasional large moves — exactly the peaked-and-fat-tailed shape a normal curve underfits. Fitting a Laplace with $b = 0.008$ (0.8% typical daily deviation) assigns a day 3 standard-deviation-equivalents out roughly 5 times the probability a normal fit would, which matches observed crash-day frequencies far better and is one reason risk models built on normal assumptions understate tail risk.