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, p. 197
Section 2.1.6.1. Distributions of sums and averages^{a}School of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and ^{b}St John's College, Cambridge, England |
In Section 2.1.2.1, it was shown that the average intensity of a sufficient number of reflections is [equation (2.1.2.4)]. When the number of reflections is not `sufficient', their mean value will show statistical fluctuations about ; such statistical fluctuations are in addition to any systematic variation resulting from non-independence of atomic positions, as discussed in Sections 2.1.2.1–2.1.2.3. We thus need to consider the probability density functions of sums like and averages like where is the intensity of the ith reflection. The probability density distributions are easily obtained from a property of gamma distributions: If are independent gamma-distributed variables with parameters , their sum is a gamma-distributed variable with parameter p equal to the sum of the parameters. The sum of n intensities drawn from an acentric distribution thus has the distribution the parameters of the variables added are all equal to unity, so that their sum is p. Similarly, the sum of n intensities drawn from a centric distribution has the distribution each parameter has the value of one-half. The corresponding distributions of the averages of n intensities are then for the acentric case, and for the centric. In both cases the expected value of Y is and the variances are and , respectively, just as would be expected.