International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B. ch. 2.1, pp. 196-197
Section 2.1.5.6. Cumulative distribution functions^{a}School of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and ^{b}St John's College, Cambridge, England |
The integral of the probability density function from the lower end of its range up to an arbitrary value x is called the cumulative probability distribution, or simply the distribution function, , of x. It can always be written if the lower end of its range is not actually one takes as identically zero between 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 ; for the distribution of , , I and the lower end of the range is zero. In such cases, equation (2.1.5.21) becomes In crystallographic applications the cumulative distribution is usually denoted by , 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 [equation (2.1.5.8)] the integration is readily carried out: The integral for the centric distribution of [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 For the centric distribution, then 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 ScholarHowells, 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