---------- *PERCENT POINT FUNCTIONS* ---------- Percent Point Functions (Inverse Cumulative Distribution Functions) for Various Distributions - - - - - - - - - - - - - - - - - - - - - - - - - Symmetric Distributions f(x) denotes probability density function G(p) denotes percent point function Uniform f(x) = 1 with x in (0,1) Uniform G(p) = 1 Normal f(x) = (1/sqrt(2pi)) * exp(-0.5*x**2) Normal G(p) not in closed form Logistic f(x) = exp(x) / (1 + exp(x))**2 Logistic G(p) = log(p/(1-p)) Double Exp. f(x) = 0.5 * exp(-x) Double Exp. G(p) = log(2p) for p <= 0.5 Double Exp. = -log(2(1-p)) for p > 0.5 Cauchy f(x) = (1/pi) * 1/(1+x**2) Cauchy G(p) = -cot(pi*p) Tukey Lambda f(x) not in closed form Tukey Lambda G(p) = (p**lambda - (1-p)**(1-lambda)) / lambda Tukey Lambda if lambda not = 0 Tukey Lambda = log(p/(1-p)) Tukey Lambda if lambda = 0 Tukey Lambda f(x) not in closed form Tukey Lambda lambda = 0.5 --U-shaped Tukey Lambda lambda = 1.0 --exactly uniform Tukey Lambda lambda = 0.14--approximately normal Tukey Lambda lambda = 0.0 --exactly logistic Tukey Lambda lambda = -1.0--approximately Cauchy Anglit f(x) = sin(2x+pi/2) with x in (-pi/4,pi/4) Anglit G(p) = arcsin(sqrt(p)) - pi/4 Triangular f(x) = 1 - abs(x) with x in (-1,1) Triangular G(p) = -1 + sqrt(2p) for p <= 0.5 Triangular = +1 - sqrt(2(1-p)) for p > 0.5 Arcsin f(x) = (1/pi) * 1 / sqrt(x(1-x)) Arcsin G(p) = (sin(pi*p/2))**2 Student t f(x) = c / [1 + (x*x/nu)]**((nu+1)/2) Student t G(p) = no simple general closed form - - - - - - - - - - - - - - - - - - - - - - - - - Skewed Distributions Chi-squared f(x) = c * [x**((nu/2)-1)] * exp(-x/2) with x >= 0 Chi-squared with x >= 0 and nu > 0 Chi-squared and where c = gamma function of nu/2 Chi-squared G(p) = no simple general closed form Gamma f(x) = c * [x**(gamma-1)] * exp(-x) with x >= 0 Gamma with x >= 0 and gamma > 0 Gamma and where c = gamma function of gamma Gamma G(p) = no simple general closed form Exponential f(x) = exp(-x) with x >= 0 Exponential G(p) = -log(1-p) Lognormal f(x) = (1/(x*sqrt(2*pi))) * exp(-0.5*(log(x))**2) Lognormal with x >= 0 Lognormal G(p) = exp(GN(p)) Lognormal where GN(p) is normal N(0,1) ppf Half-Normal f(x) = (2/sqrt(2*pi)) * exp(-0.5*x**2) Half-Normal with x >= 0 Half-Normal G(p) = GN((p+1)/2) Half-Normal where GN(p) is normal N(0,1) ppf Extreme Value I f(x) = exp(-x) * exp(-exp(-x)) Extreme Value I G(p) = -log(-log(p)) Extreme Value II f(x) = gamma * x**(-gamma-1) * exp(-(x**(-gamma)) Extreme Value II with x >= 0 and gamma > 0 Extreme Value II G(p) = (-log(p))**(-1/gamma) Weibull f(x) = gamma * x**(gamma-1) * exp(-(x**gamma)) Weibull with x > 0 and gamma > 0 Weibull G(p) = (-log(1-p))**(1/gamma) Pareto f(x) = gamma / (x**(gamma+1)) Pareto G(p) = (1-p)**(-1/gamma) Beta f(x) = c * x**(a-1) * (1-x)**(b-1) with x in (0,1) Beta where c = beta function of a and b Beta G(p) = no simple closed form - - - - - - - - - - - - - - - - - - - - - - - - - Discrete Distributions Binomial Geometric Poisson