Editing Beta distribution (section)

===Transformations===
* If ''X'' ~ Beta(''α'', ''β'') then 1 − ''X'' ~ Beta(''β'', ''α'') [[Mirror image|mirror-image]] symmetry
* If ''X'' ~ Beta(''α'', ''β'') then <math>\tfrac{X}{1-X} \sim {\beta'}(\alpha,\beta)</math>. The [[beta prime distribution]], also called "beta distribution of the second kind".
* If <math>X\sim\text{Beta}(\alpha,\beta)</math>, then <math>Y=\log\frac{X}{1-X}</math> has a [[generalized logistic distribution]], with density <math>\frac{\sigma(y)^\alpha\sigma(-y)^\beta}{B(\alpha,\beta)}</math>, where <math>\sigma</math> is the [[logistic sigmoid]].
* If ''X'' ~ Beta(''α'', ''β'') then <math>\tfrac{1}{X} -1 \sim {\beta'}(\beta,\alpha)</math>.
* If <math>X\sim\text{Beta}(\alpha_1,\beta_1)</math> and <math>Y\sim\text{Beta}(\alpha_2,\beta_2)</math> then <math>Z = \tfrac{X}{Y}</math> has density <math> \tfrac{B(\alpha_1 +\alpha_2, \beta_2) z^{\alpha_1 - 1} {}_2F_1(\alpha_1 + \alpha_2, 1- \beta_1; \alpha_1 +\alpha_2 + \beta_2; z) }{B(\alpha_1, \beta_1)B(\alpha_2, \beta_2)} </math> for <math>0 < z \leq 1 </math> and <math> \tfrac{B(\alpha_1 +\alpha_2, \beta_1) z^{-(\alpha_2 + 1)} {}_2F_1(\alpha_1 + \alpha_2, 1- \beta_2; \alpha_1 +\alpha_2 + \beta_1; \tfrac{1}{z})}{B(\alpha_1, \beta_1)B(\alpha_2, \beta_2)} </math> for <math> z \geq 1 </math>, where <math>{}_2F_1(a, b; c; x)</math> is the [[Hypergeometric function]].<ref name="Pham-Gia2000">{{cite journal |last1=Pham-Gia |first1=T. |title=Distributions of the ratios of independent beta variables and applications |journal=Communications in Statistics - Theory and Methods |date=January 2000 |volume=29 |issue=12 |pages=2693–2715 |doi=10.1080/03610920008832632 |url=https://doi.org/10.1080/03610920008832632 |access-date=13 November 2024 |language=en |issn=0361-0926}}</ref>
* If ''X'' ~ Beta(''n''/2, ''m''/2) then <math>\tfrac{mX}{n(1-X)} \sim F(n,m)</math> (assuming ''n'' > 0 and ''m'' > 0), the [[F-distribution|Fisher–Snedecor F distribution]].
* If <math>X \sim \operatorname{Beta}\left(1+\lambda\tfrac{m-\min}{\max-\min}, 1 + \lambda\tfrac{\max-m}{\max-\min}\right)</math> then min + ''X''(max − min) ~ PERT(min, max, ''m'', ''λ'') where ''PERT'' denotes a [[PERT distribution]] used in [[PERT]] analysis, and ''m''=most likely value.<ref name=NewPERT>Herrerías-Velasco, José Manuel and Herrerías-Pleguezuelo, Rafael and René van Dorp, Johan. (2011). Revisiting the PERT mean and Variance. European Journal of Operational Research (210), p. 448&ndash;451.</ref> Traditionally<ref name=Malcolm /> ''λ'' = 4 in PERT analysis.
* If ''X'' ~ Beta(1, ''β'') then ''X'' ~ [[Kumaraswamy distribution]] with parameters (1, ''β'')
* If ''X'' ~ Beta(''α'', 1) then ''X'' ~ [[Kumaraswamy distribution]] with parameters (''α'', 1)
* If ''X'' ~ Beta(''α'', 1) then −ln(''X'') ~ Exponential(''α'')