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. 198-199
Section 2.1.6.4. The use of normal approximations^{a}School of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and ^{b}St John's College, Cambridge, England |
Since and [equations (2.1.6.1) and (2.1.6.8)] 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 and 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, p. 288).
References
Kendall, M. & Stuart, A. (1977). The advanced theory of statistics, Vol. 1, 4th ed. London: Griffin.Google Scholar