// @(#)root/mathcore:$Name:  $:$Id: DistFunc.h,v 1.5 2005/12/08 21:56:39 moneta Exp $
// Authors: Andras Zsenei & Lorenzo Moneta   06/2005 

/**********************************************************************
 *                                                                    *
 * Copyright (c) 2005 , LCG ROOT MathLib Team                         *
 *                                                                    *
 *                                                                    *
 **********************************************************************/



/**

Probability density functions, cumulative distribution functions 
and their inverses of the different distributions.
Whenever possible the conventions followed are those of the
CRC Concise Encyclopedia of Mathematics, Second Edition
(or <A HREF="http://mathworld.wolfram.com/">Mathworld</A>).
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.


@author Created by Andras Zsenei on Wed Nov 17 2004

@defgroup StatFunc Statistical functions

*/






#ifndef ROOT_Math_DistFunc
#define ROOT_Math_DistFunc




namespace ROOT {
namespace Math {



  /** @name Probability Density Functions (PDF)
   *  Probability density functions of various distributions. 
   */
  //@{

  /**
    
  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 
  <A HREF="http://mathworld.wolfram.com/BinomialDistribution.html">
  Mathworld</A>. 
  
  @ingroup StatFunc

  */

  double binomial_pdf(unsigned int k, double p, unsigned int 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 StatFunc

  */

  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 
  <A HREF="http://mathworld.wolfram.com/CauchyDistribution.html">
  Mathworld</A>. It is also related to the #breitwigner_pdf which 
  will call the same implementation.
  
  @ingroup StatFunc

  */

  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 
  <A HREF="http://mathworld.wolfram.com/Chi-SquaredDistribution.html">
  Mathworld</A>. 
  
  @ingroup StatFunc

  */

  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 
  <A HREF="http://mathworld.wolfram.com/ExponentialDistribution.html">
  Mathworld</A>. 

  
  @ingroup StatFunc

  */

  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 
  <A HREF="http://mathworld.wolfram.com/F-Distribution.html">
  Mathworld</A>. 
  
  @ingroup StatFunc

  */


  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 
  <A HREF="http://mathworld.wolfram.com/GammaDistribution.html">
  Mathworld</A>. 
  
  @ingroup StatFunc

  */

  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 
  <A HREF="http://mathworld.wolfram.com/NormalDistribution.html">
  Mathworld</A>. It can also be evaluated using #normal_pdf which will 
  call the same implementation. 

  @ingroup StatFunc
 
  */

  double gaussian_pdf(double x, double sigma, double x0 = 0);



  /**
  \if later

  Probability density function of the Landau distribution.

  \f[  p(x) = \frac{1}{2 \pi i}\int_{c-i\infty}^{c+i\infty} e^{x s + s \log{s}} ds\f]

  For detailed description see 
  <A HREF="http://wwwasdoc.web.cern.ch/wwwasdoc/shortwrupsdir/g110/top.html">
  CERNLIB</A>. 
  
  @ingroup StatFunc

  \endif
  */

  // double landau_pdf(double x); 



  /**

  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 
  <A HREF="http://mathworld.wolfram.com/LogNormalDistribution.html">
  Mathworld</A>. 

  
  @ingroup StatFunc

  */

  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 
  <A HREF="http://mathworld.wolfram.com/NormalDistribution.html">
  Mathworld</A>. It can also be evaluated using #gaussian_pdf which will call the same 
  implementation. 

  @ingroup StatFunc
 
  */

  double normal_pdf(double x, double sigma, double x0 = 0);




  /**

  Probability density function of the Poisson distribution.

  \f[ p(n) = \frac{\mu^n}{n!} e^{- \mu} \f]

  For detailed description see 
  <A HREF="http://mathworld.wolfram.com/PoissonDistribution.html">
  Mathworld</A>. 
  
  @ingroup StatFunc

  */

  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 
  <A HREF="http://mathworld.wolfram.com/Studentst-Distribution.html">
  Mathworld</A>. 
  
  @ingroup StatFunc

  */

  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$a \leq x<b\f$ and 0 otherwise. For detailed description see 
  <A HREF="http://mathworld.wolfram.com/UniformDistribution.html">
  Mathworld</A>. 
  
  @ingroup StatFunc

  */

  double uniform_pdf(double x, double a, double b, double x0 = 0);




  //@}













  /**

  Multinomial distribution probability density function

  http://mathworld.wolfram.com/MultinomialDistribution.html

  */

  //double multinomial_pdf(const size_t k, const double p[], const unsigned int n[]);



} // namespace Math
} // namespace ROOT



#endif // ROOT_Math_DistFunc
