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

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

Table 2.1.5.1 

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

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)[link].

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)].]