La respuesta refleja ronda las 180, la mitad del calendario. El umbral real es 23, porque una coincidencia necesita una pareja y 23 personas cargan con 253. El mismo razonamiento sitúa la pregunta « ¿alguien comparte MI cumpleaños? » en 253 personas, once veces más gente, y el umbral en raíz cuadrada que hay detrás de ambas explica que un identificador aleatorio de 64 bits se repita tras cinco mil millones de extracciones y no tras dieciocho trillones.
También se pregunta comobirthday 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.