Wednesday, October 18, 2017

[zzfydkni] Normally distributed integers

Consider a random walk among the integers, starting at 0.  Transition in the direction +1, +0, -1 each with probability 1/3.  After a large number of steps, the endpoint is normally distributed around 0.

If you would like 95.45% (2 standard deviations) of the endpoints to be between -x and x, then take 3x^2/8 steps.  (The variance of the original distribution is 2/3, from whence the factor of 3.)

150 steps yields normally distributed numbers usually between -20 and 20.  15000 yields -200 to 200.  (We artificially induce Benford's Law: an initial digit of 1 seems more natural.)

No comments :