Normal Probability Function

[Albert Chan]

(03-29-2022 12:00 PM)Paul Dale Wrote:  Upper Tail = 0.5*gammaq(0.5*x^2, 0.5)

Correction, factor off by 1/√(pi)

Γ(1/2, x²) = √(pi) * erfc(x)      // Abramowitz and Stegun, 6.5.17

P(x) = erfc(x/√2)/2 = Γ(1/2, x²/2) / (2*√(pi))

>>> from mpmath import *
>>> P1 = lambda x: erfc(x/sqrt(2))/2
>>> P2 = lambda x: gammainc(1/2,x*x/2) / (2*sqrt(pi))
>>> z = 1
>>> print P1(z), P2(z) # confirm P1 = P2
0.158655253931457 0.158655253931457

Quote:The common approach is to break the domain into two pieces and compute the function
using a series in one part and a continued fraction for the other.

For Xcas implementation of erf, see https://www.hpmuseum.org/forum/thread-15...#pid135589