// @(#)root/mathcore:$Id$ // Authors: Andras Zsenei & Lorenzo Moneta 06/2005 /********************************************************************** * * * Copyright (c) 2005 , LCG ROOT MathLib Team * * * * * **********************************************************************/ /** Probability density functions, cumulative distribution functions and their inverses (quantiles) for various statistical distributions (continuous and discrete). Whenever possible the conventions followed are those of the CRC Concise Encyclopedia of Mathematics, Second Edition (or Mathworld). By convention the distributions are centered around 0, so for example in the case of a Gaussian there is no parameter mu. The user must calculate the shift himself if he wishes. MathCore provides the majority of the probability density functions, of the cumulative distributions and of the quantiles (inverses of the cumulatives). Additional distributions are also provided by the MathMore library. @defgroup StatFunc Statistical functions */ #if defined(__CINT__) && !defined(__MAKECINT__) // avoid to include header file when using CINT #ifndef _WIN32 #include "../lib/libMathCore.so" #else #include "../bin/libMathCore.dll" #endif #else #ifndef ROOT_Math_PdfFuncMathCore #define ROOT_Math_PdfFuncMathCore namespace ROOT { namespace Math { /** @defgroup PdfFunc Probability Density Functions (PDF) from MathCore * @ingroup StatFunc * Probability density functions of various statistical distributions * (continuous and discrete). * The probability density function returns the probability that * the variate has the value x. * In statistics the PDF is also called the frequency function. * * */ /** @name Probability Density Functions from MathCore * Additional PDF's are provided in the MathMore library * (see PDF functions from MathMore) */ //@{ /** Probability density function of the beta distribution. \f[ p(x) = \frac{\Gamma (a + b) } {\Gamma(a)\Gamma(b) } x ^{a-1} (1 - x)^{b-1} \f] for \f$0 \leq x \leq 1 \f$. For detailed description see Mathworld. @ingroup PdfFunc */ double beta_pdf(double x, double a, double b); /** Probability density function of the binomial distribution. \f[ p(k) = \frac{n!}{k! (n-k)!} p^k (1-p)^{n-k} \f] for \f$ 0 \leq k \leq n \f$. For detailed description see Mathworld. @ingroup PdfFunc */ double binomial_pdf(unsigned int k, double p, unsigned int n); /** Probability density function of the negative binomial distribution. \f[ p(k) = \frac{(k+n-1)!}{k! (n-1)!} p^{n} (1-p)^{k} \f] For detailed description see Mathworld (where $k \to x$ and $n \to r$). The distribution in Wikipedia is defined with a $p$ corresponding to $1-p$ in this case. @ingroup PdfFunc */ double negative_binomial_pdf(unsigned int k, double p, double n); /** Probability density function of Breit-Wigner distribution, which is similar, just a different definition of the parameters, to the Cauchy distribution (see #cauchy_pdf ) \f[ p(x) = \frac{1}{\pi} \frac{\frac{1}{2} \Gamma}{x^2 + (\frac{1}{2} \Gamma)^2} \f] @ingroup PdfFunc */ double breitwigner_pdf(double x, double gamma, double x0 = 0); /** Probability density function of the Cauchy distribution which is also called Lorentzian distribution. \f[ p(x) = \frac{1}{\pi} \frac{ b }{ (x-m)^2 + b^2} \f] For detailed description see Mathworld. It is also related to the #breitwigner_pdf which will call the same implementation. @ingroup PdfFunc */ double cauchy_pdf(double x, double b = 1, double x0 = 0); /** Probability density function of the \f$\chi^2\f$ distribution with \f$r\f$ degrees of freedom. \f[ p_r(x) = \frac{1}{\Gamma(r/2) 2^{r/2}} x^{r/2-1} e^{-x/2} \f] for \f$x \geq 0\f$. For detailed description see Mathworld. @ingroup PdfFunc */ double chisquared_pdf(double x, double r, double x0 = 0); /** Probability density function of the exponential distribution. \f[ p(x) = \lambda e^{-\lambda x} \f] for x>0. For detailed description see Mathworld. @ingroup PdfFunc */ double exponential_pdf(double x, double lambda, double x0 = 0); /** Probability density function of the F-distribution. \f[ p_{n,m}(x) = \frac{\Gamma(\frac{n+m}{2})}{\Gamma(\frac{n}{2}) \Gamma(\frac{m}{2})} n^{n/2} m^{m/2} x^{n/2 -1} (m+nx)^{-(n+m)/2} \f] for x>=0. For detailed description see Mathworld. @ingroup PdfFunc */ double fdistribution_pdf(double x, double n, double m, double x0 = 0); /** Probability density function of the gamma distribution. \f[ p(x) = {1 \over \Gamma(\alpha) \theta^{\alpha}} x^{\alpha-1} e^{-x/\theta} \f] for x>0. For detailed description see Mathworld. @ingroup PdfFunc */ double gamma_pdf(double x, double alpha, double theta, double x0 = 0); /** Probability density function of the normal (Gaussian) distribution. \f[ p(x) = {1 \over \sqrt{2 \pi \sigma^2}} e^{-x^2 / 2\sigma^2} \f] For detailed description see Mathworld. It can also be evaluated using #normal_pdf which will call the same implementation. @ingroup PdfFunc */ double gaussian_pdf(double x, double sigma = 1, double x0 = 0); /** Probability density function of the Landau distribution: \f[ p(x) = \frac{1}{\xi} \phi (\lambda) \f] with \f[ \phi(\lambda) = \frac{1}{2 \pi i}\int_{c-i\infty}^{c+i\infty} e^{\lambda s + s \log{s}} ds\f] where \f$\lambda = (x-x_0)/\xi\f$. For a detailed description see K.S. Kölbig and B. Schorr, A program package for the Landau distribution, Computer Phys. Comm. 31 (1984) 97-111 [Erratum-ibid. 178 (2008) 972]. The same algorithms as in CERNLIB (DENLAN) is used @param x The argument \f$x\f$ @param xi The width parameter \f$\xi\f$ @param x0 The location parameter \f$x_0\f$ @ingroup PdfFunc */ double landau_pdf(double x, double xi = 1, double x0 = 0); /** Probability density function of the lognormal distribution. \f[ p(x) = {1 \over x \sqrt{2 \pi s^2} } e^{-(\ln{x} - m)^2/2 s^2} \f] for x>0. For detailed description see Mathworld. @param s scale parameter (not the sigma of the distribution which is not even defined) @param x0 location parameter, corresponds approximatly to the most probable value. For x0 = 0, sigma = 1, the x_mpv = -0.22278 @ingroup PdfFunc */ double lognormal_pdf(double x, double m, double s, double x0 = 0); /** Probability density function of the normal (Gaussian) distribution. \f[ p(x) = {1 \over \sqrt{2 \pi \sigma^2}} e^{-x^2 / 2\sigma^2} \f] For detailed description see Mathworld. It can also be evaluated using #gaussian_pdf which will call the same implementation. @ingroup PdfFunc */ double normal_pdf(double x, double sigma =1, double x0 = 0); /** Probability density function of the Poisson distribution. \f[ p(n) = \frac{\mu^n}{n!} e^{- \mu} \f] For detailed description see Mathworld. @ingroup PdfFunc */ double poisson_pdf(unsigned int n, double mu); /** Probability density function of Student's t-distribution. \f[ p_{r}(x) = \frac{\Gamma(\frac{r+1}{2})}{\sqrt{r \pi}\Gamma(\frac{r}{2})} \left( 1+\frac{x^2}{r}\right)^{-(r+1)/2} \f] for \f$k \geq 0\f$. For detailed description see Mathworld. @ingroup PdfFunc */ double tdistribution_pdf(double x, double r, double x0 = 0); /** Probability density function of the uniform (flat) distribution. \f[ p(x) = {1 \over (b-a)} \f] if \f$a \leq x Mathworld. @ingroup PdfFunc */ double uniform_pdf(double x, double a, double b, double x0 = 0); } // namespace Math } // namespace ROOT #endif // ROOT_Math_PdfFunc #endif // if defined (__CINT__) && !defined(__MAKECINT__)