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

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

Section 2.1.6.1. Distributions of sums and averages

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

2.1.6.1. Distributions of sums and averages

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In Section 2.1.2.1[link], it was shown that the average intensity of a sufficient number of reflections is [\Sigma] [equation (2.1.2.4)[link]]. When the number of reflections is not `sufficient', their mean value will show statistical fluctuations about [\Sigma]; 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[link][link][link]. We thus need to consider the probability density functions of sums like [J_{n} = \textstyle\sum\limits_{i = 1}^{n}G_{i}, \eqno(2.1.6.1)] and averages like [Y = J_{n}/n, \eqno(2.1.6.2)] where [G_{i}] is the intensity of the ith reflection. The probability density distributions are easily obtained from a property of gamma distributions: If [x_{1}, x_{2}, \ldots, x_{n}] are independent gamma-distributed variables with parameters [p_{1}, p_{2}, \ldots, p_{n}], 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 [p(J_{n})\;{\rm d}J_{n} = \gamma_{n}(J_{n}/\Sigma)\;{\rm d}(J_{n}/\Sigma)\hbox{;} \eqno(2.1.6.3)] 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 [p(J_{n})\;{\rm d}J_{n} = \gamma_{n/2}[J_{n}/(2\Sigma)]\;{\rm d}[J_{n}/(2\Sigma)]\hbox{;} \eqno(2.1.6.4)] each parameter has the value of one-half. The corresponding distributions of the averages of n intensities are then [p(Y)\;{\rm d}Y = \gamma_{n}(nY/\Sigma)\;{\rm d}(nY/\Sigma) \eqno(2.1.6.5)] for the acentric case, and [p(Y)\;{\rm d}Y = \gamma_{n/2}[nY/(2\Sigma)]\;{\rm d}[nY/(2\Sigma)] \eqno(2.1.6.6)] for the centric. In both cases the expected value of Y is [\Sigma] and the variances are [\Sigma^{2}/n] and [2\Sigma^{2}/n], respectively, just as would be expected.








































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