Probability — page 5
When to stop tossing a coin
Toss a fair coin as many times as you like and stop whenever you choose; you are paid the proportion of heads so far. Stopping after a first head pays one, so the only question is what to do after a tail. Waiting for the first moment heads lead pays exactly π/4 on average. Waiting for a lead of two pays exactly ln 2. The best possible rule pays about 0.79295, a number known to six decimal places only since 2013 — and it stops when the lead reaches about 0.84 times the square root of the number of tosses, a constant that comes from Brownian motion.
The middle value has a bell of its own
The central limit theorem is a theorem about sums, and the median of a sample is not a sum of anything. It has a bell all the same: the middle of n draws spreads like 1/(2f√n), where f is the height of the density at the middle — not the variance, which the median never consults. That one number decides when the median beats the average, by a factor of two for a two-sided exponential and outright for the Cauchy law; and where the density is nought at the middle, the bell fails, slowing to a fourth root or splitting in two.