(15C)(DM15) - PDF/CDF and Inverse of a Normal Distribution

[deetee]

(11-28-2021 05:37 PM)Albert Chan Wrote:  → z ≈ 5.494 * sinh(asinh(-0.3418*ln(1/CDF-1))/3)

Hello Albert!

Thanks for your formula - but I have problems because CDF has to be between 0 and 1 - and CDF-1 is negative, where the ln ist not defined ... hmm (or am I wrong?).

But inspired by your "jaunty and cool " math-handling (and the TV-News from yesterday, where COVID-scientists worked with PDF and CDF functions) I developed some (new) formulae by myself (using hyperbolic functions):

CDF(x) ≈ (1 + tanh(7/200* x^3 + 4/5 * x)) / 2 ... Maximal error < 4.10-4

And the corresponding Inverse-CDF (one real root of this cubic equation):

x(CDF) = a * sinh(1/3 * asinh(b * atanh(1 - 2*CDF)))

where a = -2 * sqrt(160/7/3) ≈ -5.52
and b = 3/32 * sqrt(7*15/2) ≈ 0.679

Probably I don't change my original HP-15C-program because of many more program steps for a little bit faster calculation (no SOLVE) but for my standard linux console calculator (CLAC, FORTH-like, no SOLVE-function) it's worth to define new functions for PDF, CDF and their inverse.

Regards
deetee