---------- *PROBABILITY DENSITY FUNCTIONS* ---------- Probability Density Functions for Various Distributions - - - - - - - - - - - - - - - - - - - - - - - - - Symmetric Distributions Uniform f(x) = 1 with x in (0,1) Normal f(x) = (1/sqrt(2pi)) * exp(-0.5*x**2) Logistic f(x) = exp(x) / (1 + exp(x))**2 Double Exp. f(x) = 0.5 * exp(-x) Cauchy f(x) = (1/pi) * 1/(1+x**2) 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) Triangular f(x) = 1 - abs(x) with x in (-1,1) Arcsin f(x) = (1/pi) * 1 / sqrt(x(1-x)) Student t f(x) = c / [1 + (x*x/nu)]**((nu+1)/2) - - - - - - - - - - - - - - - - - - - - - - - - - Skewed Distributions Chi-squared f(x) = c * [x**((nu/2)-1)] * exp(-x/2) Chi-squared with x >= 0 and nu > 0 Chi-squared and where c = gamma function of nu/2 Gamma f(x) = c * [x**(gamma-1)] * exp(-x) Gamma with x >= 0 and gamma > 0 Gamma and where c = gamma function of gamma Exponential f(x) = exp(-x) with x >= 0 Lognormal f(x) = (1/(x*sqrt(2*pi))) * exp(-0.5*(log(x))**2) Lognormal with x >= 0 Half-Normal f(x) = (2/sqrt(2*pi)) * exp(-0.5*x**2) Half-Normal with x >= 0 Half-Normal where GN(p) is normal N(0,1) ppf Extreme Value I f(x) = exp(-x) * exp(-exp(-x)) Extreme Value II f(x) = gamma * x**(-gamma-1) * exp(-(x**(-gamma)) Extreme Value II with x >= 0 and gamma > 0 Weibull f(x) = gamma * x**(gamma-1) * exp(-(x**gamma)) Weibull with x > 0 and gamma > 0 Pareto f(x) = gamma / (x**(gamma+1)) 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 - - - - - - - - - - - - - - - - - - - - - - - - - Discrete Distributions Binomial Geometric Poisson