Table 2.1.5.1

Shmueli, U.; Wilson, A. J. C.

doi:10.1107/97809553602060000554

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

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

Table 2.1.5.1

U. Shmueli^a ^* and A. J. C. Wilson^b ^‡

^a School 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

Table 2.1.5.1 | top | pdf |
Some properties of gamma and beta 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 $[p\ =]$ $[ p_{1} + p_{2} + \ldots + p_{n}]$ .

If x and y are independent gamma-distributed variables with parameters p and q, then the ratio [u = x/y] has the distribution $[\beta_{2} (u\hbox{; } p, q)]$ .

With the same notation, the ratio [v = x/(x + y)] has the distribution $[\beta_{1} (v\hbox{; }p, q)]$ .

Differences and products of gamma-distributed variables do not lead to simple results. For proofs, details and references see Kendall & Stuart (1977).

Name of the distribution, its functional form, mean and variance
Gamma distribution with parameter p: $[\gamma_{p} (x) = [\Gamma (x)]^{-1} x^{p-1} \exp (-x)\hbox{;} \quad p \leq x \leq \infty,\quad p > 0]$ $[\hbox{mean: }\langle x\rangle = p\hbox{;} \quad \hbox{variance: } \langle (x - \langle x\rangle)^{2}\rangle = p.]$
Beta distribution of first kind with parameters p and q: $[\beta_{1} (x\hbox{; } p, q) = {\Gamma (p + q) \over \Gamma (p) \Gamma (q)} x^{p - 1} (1 - x)^{q - 1}\hbox{;} \quad 0 \leq x \leq \infty,\quad p, q > 0]$ $[\hbox{mean: }\langle x\rangle = p/(p + q)\hbox{;}]$ $[\hbox{variance: }\langle (x - \langle x \rangle)^{2}\rangle = pq/[(p + q)^{2} (p + q + 1)].]$
Beta distribution of second kind with parameters p and q: $[\beta_{2} (x\hbox{; } p, q) = {\Gamma (p + q) \over \Gamma (p) \Gamma (q)} x^{p - 1} (1 + x)^{-p -q}\hbox{;} \quad 0 \leq x \leq \infty,\quad p, q > 0]$ $[\hbox{mean: }\langle x \rangle = p/(q - 1);]$ $[\hbox{variance: }\langle (x - \langle x \rangle)^{2}\rangle = p(p + q - 1)/[(q - 1) (q - 2)].]$