Coincidentally, I have been wondering about just how closely the (scaled) Gaussian curve approaches the (discrete) binomial coefficients; this involves inverting the standard investigation in (say) Feller vol I, chap. VII (2.11) p.170 to inspect horizontal rather than vertical distance between the curves. With y = exp(-x^2/2) / sqrt(2\pi) the unit Gaussian function, easily h := 1/sqrt(n/4); y := binomial(n, k)/(2^n*h); x := sign(k - n/2)*sqrt(-2*log(y*sqrt(2*Pi))); k' := x/h + n/2; now experimentally it appears that |k - k'| < 1/2 , except near endpoints k < O(sqrt(n) etc. So what I want is some strict bound on |k - k'| --- maybe well-known? Incidentally, I have a note in my copy of Feller that his analysis may be incomplete, with a reference to Y. S. Chow & H. Teicher "Probability Theory" Springer (1978), sect 2.3 Lemma 2 . Can anybody comment? Fred Lunnon On 7/9/14, Mike Stay <metaweta@gmail.com> wrote:
(sqrt(pi) x)/2 + 1/24 pi^(3/2) x^3 + 7/960 pi^(5/2) x^5 + O(x^7)
http://www.wolframalpha.com/input/?i=taylor+series+of+InverseErf
On Tue, Jul 8, 2014 at 10:24 PM, Warren D Smith <warren.wds@gmail.com> wrote:
If x=erf(y), then say y=erfinv(x).
SMP jocks: What is the Maclaurin series of erfinv(x)?
Obviously the erfinv function is useful for statistics. It also occurs to me: This series should have infinite radius of convergence, i.e. erfinv(x) should be a very well behaved "entire" function. Because: the derivative of erf is never zero anywhere in the complex plane, and the value of erf is always finite everywhere in the plane, so its inverse function should exist everywhere.
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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