Cumulative distribution functions

Shmueli, U.; Wilson, A. J. C.

doi:10.1107/97809553602060000554

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 2.1, pp. 196-197 | 1 | 2 |

Section 2.1.5.6. Cumulative distribution functions

U. Shmueli^a ^* and A. J. C. Wilson^b ^‡

^a School of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and ^bSt John's College, Cambridge, England
Correspondence e-mail: ushmueli@post.tau.ac.il

2.1.5.6. Cumulative distribution functions

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The integral of the probability density function [f(x)] from the lower end of its range up to an arbitrary value x is called the cumulative probability distribution, or simply the distribution function, [F(x)] , of x. It can always be written $[F(x) = \textstyle\int\limits_{-\infty}^{x}f(u)\;{\rm d}u\hbox{;} \eqno(2.1.5.21)]$ if the lower end of its range is not actually $[-\infty]$ one takes [f(x)] as identically zero between $[-\infty]$ and the lower end of its range. For the distribution of A [equation (2.1.5.4) or (2.1.5.9)] the lower limit is in fact $[-\infty]$ ; for the distribution of [|F|] , [|E|] , I and $[I/\Sigma]$ the lower end of the range is zero. In such cases, equation (2.1.5.21) becomes $[F(x) = \textstyle\int\limits_{0}^{x}f(x)\;{\rm d}x. \eqno(2.1.5.22)]$ In crystallographic applications the cumulative distribution is usually denoted by [N(x)] , rather than by the capital letter corresponding to the probability density function designation. The cumulative forms of the ideal acentric and centric distributions (Howells et al., 1950) have found many applications. For the acentric distribution of [|E|] [equation (2.1.5.8)] the integration is readily carried out: $[N(|E|) = 2\textstyle\int\limits_{0}^{|E|} y\exp(-y^{2})\;{\rm d}y = 1 - \exp(-|E|^{2}). \eqno(2.1.5.23)]$ The integral for the centric distribution of [|E|] [equation (2.1.5.11)] cannot be expressed in terms of elementary functions, but the integral required has so many important applications in statistics that it has been given a special name and symbol, the error function erf(x), defined by $[{\rm erf}(x) = (2/\pi^{1/2})\textstyle\int\limits_{0}^{x}\exp(-t^{2})\;{\rm d}t. \eqno(2.1.5.24)]$ For the centric distribution, then $[\eqalignno{N(|E|) & = (2/\pi)^{1/2}\textstyle\int\limits_{0}^{|E|}y\exp(-y^{2}/2)\;{\rm d}y &(2.1.5.25) \cr & = {\rm erf}(|E|/2^{1/2}). &(2.1.5.26)}%2.1.5.26]$ The error function is extensively tabulated [see e.g. Abramowitz & Stegun (1972), pp. 310–311, and a closely related function on pp. 966–973].

References

Abramowitz, M. & Stegun, I. A. (1972). Handbook of mathematical functions. New York: Dover.Google Scholar

Howells, E. R., Phillips, D. C. & Rogers, D. (1950). The probability distribution of X-ray intensities. II. Experimental investigation and the X-ray detection of centers of symmetry. Acta Cryst. 3, 210–214.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 2.1, pp. 196-197