The direct way provides a normal standard r.v. z from a chi-square y with r degrees of freedom: z = [(y/r)^(1/3) - (1- 2/(9*r)]/sqrt(2/(9*r)) Inversely, z*sqrt(2/(9*r)) + (1-2/(9*r)) = (y/r)^(1/3) _______ Corollary: With s=2/(9*r) we get y= r*[z*sqrt(s) + (1-s)]^3 ___(1) y being a chi-square r.v. obtained from a z~N(0,1).
Numerical Example With z= 1, r= 5 s=0.0444. . . y= 7.9338 (from Kalkulator) Then p(chi,5df < y) = 0.8401 In fact p(z< 1) = 0.8413 is the exact value.
A direct application, which avoids the simulation of the samples, is that we can obtain the results from difference of normal sample means, same variance, by the C.I. semi-amplitude ssdX ,ssdY, the sum of squared deviations, +/- t(nX+ nY-2) * k * sqrt(ssdX+ ssdY) Here k= sqrt[1/(nX+ nY- 2)*(1/nX+ 1/nY)] Because the quantity ssdX+ ssdY have a Chi-square distribution, nX+ nY- 2 df, the result, although approximate, do follow demands only one step.