La réponse réflexe tourne autour de 180, la moitié du calendrier. Le vrai seuil est 23, parce qu'une coïncidence demande une paire et que 23 personnes en portent 253. Le même raisonnement met la question « quelqu'un partage-t-il MON anniversaire » à 253 personnes, onze fois plus de monde, et le seuil en racine carrée derrière les deux explique qu'un identifiant aléatoire de 64 bits se répète après cinq milliards de tirages et non dix-huit trillions.
Aussi posé sous le nom debirthday problem · birthday paradox · shared birthday probability · 23 people birthday
A plain t sitting next to the t squared in a Gaussian exponent looks like a new function and is only a shift. Completing the square turns the integral of e to the minus a t squared over two plus b t, from x to infinity, into e to the b squared over 2a times the root of 2 pi over a times the standard normal at a rescaled and shifted argument, never at x itself. The worked case comes out as exactly half a bell, e root pi over two or 2.40901455, but only because its lower limit happens to land on the centre b over a.
Daily, weekly and monthly returns give per-day variance estimates of 1.0000, 1.3225 and 1.4000, and the reflex is to average them into 1.2408, a figure no horizon produced. The variance ratio is a weighted sum of autocorrelations, so a forty percent overshoot at twenty periods measures dependence rather than noise, and the coefficient that reproduces it is 0.17554. With twenty years of daily data that ratio sits 4.6 standard errors above one and with five years only 2.3, which is why the number means nothing without the sample size attached.
Two weights that add to 0.98 give the variance forecast a half-life of 34.31 days; delete the second one and the half-life is 0.2744 days, gone before the next open. The same recursion turns strictly normal daily draws into a year with kurtosis exactly 297/67, and one shuffle of those same numbers separates the fat tail from the clustering. It also has a condition nobody quotes: stationarity is not enough for that kurtosis to be finite.