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

International Tables for Crystallography (2006). Vol. B. ch. 2.1, pp. 198-199   | 1 | 2 |

Section 2.1.6.4. The use of normal approximations

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

2.1.6.4. The use of normal approximations

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Since [J_{n}] and [K_{m}] [equations (2.1.6.1)[link] and (2.1.6.8)[link]] are sums of identically distributed variables conforming to the conditions of the central-limit theorem, it is tempting to approximate their distributions by normal distributions with the correct mean and variance. This would be reasonably satisfactory for the distributions of [J_{n}] and [K_{m}] themselves for quite small values of n and m, but unsatisfactory for the distribution of their ratio for any values of n and m, even large. The ratio of two variables with normal distributions is notorious for its rather indeterminate mean and `infinite' variance, resulting from the `tail' of the denominator distributions extending through zero to negative values. The leading terms of the ratio distribution are given by Kendall & Stuart (1977[link], p. 288).

References

First citationKendall, M. & Stuart, A. (1977). The advanced theory of statistics, Vol. 1, 4th ed. London: Griffin.Google Scholar








































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