Question

How can I set points on a 24h period spreaded by the Gaussian distributions? For example to have the peak at 10 o’clock?

Answer 1

The following code generates a gaussian distributed random time (in hours, plus fractions of an hour) centered at a given time, and with a given standard deviation. The random times may ‘wrap around’ the clock, especially if the standard deviation is several hours: this is handled correctly. A different ‘wrapping’ algorithm may be more efficient if your standard deviations are very large (many days), but the distribution will be almost uniform in this case, anyway.

$peak=10; // Peak at 10-o-clock $stdev=2; // Standard deviation of two hours $hoursOnClock=24; // 24-hour clock  do // Generate gaussian variable using Box-Muller {     $u=2.0*mt_rand()/mt_getrandmax()-1.0;     $v=2.0*mt_rand()/mt_getrandmax()-1.0;     $s = $u*$u+$v*$v; } while ($s > 1); $gauss=$u*sqrt(-2.0*log($s)/$s);  $gauss = $gauss*$stdev + $peak; // Transform to correct peak and standard deviation  while ($gauss < 0) $gauss+=$hoursOnClock; // Wrap around hours to keep the random time  $result = fmod($gauss,$hoursOnClock);     // on the clock  echo $result;