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Re: Question regarding claimed approximation to standard normal CDF
Posted:
Nov 10, 2011 6:48 AM


ResentFrom: <bergv@illinois.edu> From: Dan Luecking <LookInSig@uark.edu> Date: November 9, 2011 12:29:50 PM MST To: "scimathresearch@moderators.isc.org" <scimathresearch@moderators.isc.org> Subject: Re: Question regarding claimed approximation to standard normal CDF
On Wed, 2 Nov 2011 16:58:37 +0000, jdm <james.d.mclaughlin@gmail.com> wrote:
> I recently encountered the following claim in a cryptography paper > regarding the CDF of the standard normal distribution: > > Phi(b) \approx e^{b^{2}/2}/sqrt(2*pi) when b is large. > > This looks almost exactly the same as the exact formula, the > difference being that the integration db is omitted. > > I have been unable to find any other source for this approximation, > and when I emailed the author I was referred to another paper which > used the approximation without providing evidence of its validity. > > Could someone please supply me with a source for this approximation, > such as a textbook in which it appears, or, if it is in fact not > valid, tell me so?
It depends on what \approx means. In fact b \Phi(b) < e^{b^{2}/2} / \sqrt{2*pi} for all positive b.
This useful upper bound on Phi(b) can be obtained as follows. For x \le b < 0 we get b \le x and so
b \Phi(b) = \int_{\infty}^{b} b e^{x^2/2} dx / \sqrt{2\pi} < \int_{\infty}^{b} x e^{x^2/2} dx / \sqrt{2\pi} = e^{b^2/2} / \sqrt{2\pi}
Dan To reply by email, change LookInSig to luecking



