Normal Probability Function

[Paul Dale]

These functions are very difficult to compute accurately over their entire domain. The WP 34S uses the incomplete gamma function to work them out:

Lower Tail = 0.5*(1 + gammap(0.5*x^2, 0.5))
Upper Tail = 0.5*gammaq(0.5*x^2, 0.5)

This belies the difficulty computing gammap and gammaq accurately. 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. See e.g. section 6.5 in Abramowitz and Stegun. This fails for large arguments from memory and more recent approaches break the domain into more pieces. All this relies on being able to compute the (standard) gamma function accurately which is another problematic one -- albeit nowhere near as nasty as the imcomplete gamma function.

If you do implement the incomplte gamma function, you also get Poisson and χ² distributions for free.


Pauli