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

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

Section 2.1.5.6. Cumulative distribution functions

U. Shmuelia* and A. J. C. Wilsonb

aSchool 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)[link] or (2.1.5.9)[link]] 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)[link] 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[link]) have found many applications. For the acentric distribution of [|E|] [equation (2.1.5.8)[link]] 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)[link]] 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[link]), pp. 310–311, and a closely related function on pp. 966–973].

References

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








































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