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Cumulative Distribution Functions (CDF)

Cumulative distribution functions of various distributions.

The functions with the extension _cdf calculate the lower tail integral of the probability density function

$D(x) = \int_{-\infty}^{x} p(x') dx'$

while those with the _cdf_c extension calculate the complement of cumulative distribution function, called in statistics the survival function. It corresponds to the upper tail integral of the probability density function

$D(x) = \int_{x}^{+\infty} p(x') dx'$

NOTE: In the old releases (< 5.14) the _cdf functions were called _quant and the _cdf_c functions were called _prob. These names are currently kept for backward compatibility, but their usage is deprecated.

These functions are defined in the header file Math/ProbFunc.h or in the global one including all statistical functions Math/DistFunc.h

## Functions

double ROOT::Math::beta_cdf (double x, double a, double b)
Cumulative distribution function of the beta distribution Upper tail of the integral of the beta_pdf.

double ROOT::Math::beta_cdf_c (double x, double a, double b)
Complement of the cumulative distribution function of the beta distribution.

double ROOT::Math::binomial_cdf (unsigned int k, double p, unsigned int n)
Cumulative distribution function of the Binomial distribution Lower tail of the integral of the binomial_pdf.

double ROOT::Math::binomial_cdf_c (unsigned int k, double p, unsigned int n)
Complement of the cumulative distribution function of the Binomial distribution.

double ROOT::Math::breitwigner_cdf (double x, double gamma, double x0=0)
Cumulative distribution function (lower tail) of the Breit_Wigner distribution and it is similar (just a different parameter definition) to the Cauchy distribution (see cauchy_cdf )

double ROOT::Math::breitwigner_cdf_c (double x, double gamma, double x0=0)
Complement of the cumulative distribution function (upper tail) of the Breit_Wigner distribution and it is similar (just a different parameter definition) to the Cauchy distribution (see cauchy_cdf_c )

double ROOT::Math::cauchy_cdf (double x, double b, double x0=0)
Cumulative distribution function (lower tail) of the Cauchy distribution which is also Lorentzian distribution.

double ROOT::Math::cauchy_cdf_c (double x, double b, double x0=0)
Complement of the cumulative distribution function (upper tail) of the Cauchy distribution which is also Lorentzian distribution.

double ROOT::Math::chisquared_cdf (double x, double r, double x0=0)
Cumulative distribution function of the $$\chi^2$$ distribution with $$r$$ degrees of freedom (lower tail).

double ROOT::Math::chisquared_cdf_c (double x, double r, double x0=0)
Complement of the cumulative distribution function of the $$\chi^2$$ distribution with $$r$$ degrees of freedom (upper tail).

double ROOT::Math::crystalball_cdf (double x, double alpha, double n, double sigma, double x0=0)
Cumulative distribution for the Crystal Ball distribution function.

double ROOT::Math::crystalball_cdf_c (double x, double alpha, double n, double sigma, double x0=0)
Complement of the Cumulative distribution for the Crystal Ball distribution.

double ROOT::Math::crystalball_integral (double x, double alpha, double n, double sigma, double x0=0)
Integral of the not-normalized Crystal Ball function.

double ROOT::Math::exponential_cdf (double x, double lambda, double x0=0)
Cumulative distribution function of the exponential distribution (lower tail).

double ROOT::Math::exponential_cdf_c (double x, double lambda, double x0=0)
Complement of the cumulative distribution function of the exponential distribution (upper tail).

double ROOT::Math::fdistribution_cdf (double x, double n, double m, double x0=0)
Cumulative distribution function of the F-distribution (lower tail).

double ROOT::Math::fdistribution_cdf_c (double x, double n, double m, double x0=0)
Complement of the cumulative distribution function of the F-distribution (upper tail).

double ROOT::Math::gamma_cdf (double x, double alpha, double theta, double x0=0)
Cumulative distribution function of the gamma distribution (lower tail).

double ROOT::Math::gamma_cdf_c (double x, double alpha, double theta, double x0=0)
Complement of the cumulative distribution function of the gamma distribution (upper tail).

double ROOT::Math::landau_cdf (double x, double xi=1, double x0=0)
Cumulative distribution function of the Landau distribution (lower tail).

double ROOT::Math::landau_cdf_c (double x, double xi=1, double x0=0)
Complement of the distribution function of the Landau distribution (upper tail).

double ROOT::Math::lognormal_cdf (double x, double m, double s, double x0=0)
Cumulative distribution function of the lognormal distribution (lower tail).

double ROOT::Math::lognormal_cdf_c (double x, double m, double s, double x0=0)
Complement of the cumulative distribution function of the lognormal distribution (upper tail).

double ROOT::Math::negative_binomial_cdf (unsigned int k, double p, double n)
Cumulative distribution function of the Negative Binomial distribution Lower tail of the integral of the negative_binomial_pdf.

double ROOT::Math::negative_binomial_cdf_c (unsigned int k, double p, double n)
Complement of the cumulative distribution function of the Negative Binomial distribution.

double ROOT::Math::normal_cdf (double x, double sigma=1, double x0=0)
Cumulative distribution function of the normal (Gaussian) distribution (lower tail).

double ROOT::Math::normal_cdf_c (double x, double sigma=1, double x0=0)
Complement of the cumulative distribution function of the normal (Gaussian) distribution (upper tail).

double ROOT::Math::poisson_cdf (unsigned int n, double mu)
Cumulative distribution function of the Poisson distribution Lower tail of the integral of the poisson_pdf.

double ROOT::Math::poisson_cdf_c (unsigned int n, double mu)
Complement of the cumulative distribution function of the Poisson distribution.

double ROOT::Math::tdistribution_cdf (double x, double r, double x0=0)
Cumulative distribution function of Student's t-distribution (lower tail).

double ROOT::Math::tdistribution_cdf_c (double x, double r, double x0=0)
Complement of the cumulative distribution function of Student's t-distribution (upper tail).

double ROOT::Math::uniform_cdf (double x, double a, double b, double x0=0)
Cumulative distribution function of the uniform (flat) distribution (lower tail).

double ROOT::Math::uniform_cdf_c (double x, double a, double b, double x0=0)
Complement of the cumulative distribution function of the uniform (flat) distribution (upper tail).

double ROOT::Math::vavilov_accurate_cdf (double x, double kappa, double beta2)
The Vavilov cumulative probability density function.

double ROOT::Math::vavilov_accurate_cdf_c (double x, double kappa, double beta2)
The Vavilov complementary cumulative probability density function.

double ROOT::Math::vavilov_fast_cdf (double x, double kappa, double beta2)
The Vavilov cumulative probability density function.

double ROOT::Math::vavilov_fast_cdf_c (double x, double kappa, double beta2)
The Vavilov complementary cumulative probability density function.

## ◆ beta_cdf()

 double ROOT::Math::beta_cdf ( double x, double a, double b )

Cumulative distribution function of the beta distribution Upper tail of the integral of the beta_pdf.

Definition at line 27 of file ProbFuncMathCore.cxx.

## ◆ beta_cdf_c()

 double ROOT::Math::beta_cdf_c ( double x, double a, double b )

Complement of the cumulative distribution function of the beta distribution.

Upper tail of the integral of the beta_pdf

Definition at line 20 of file ProbFuncMathCore.cxx.

## ◆ binomial_cdf()

 double ROOT::Math::binomial_cdf ( unsigned int k, double p, unsigned int n )

Cumulative distribution function of the Binomial distribution Lower tail of the integral of the binomial_pdf.

Definition at line 304 of file ProbFuncMathCore.cxx.

## ◆ binomial_cdf_c()

 double ROOT::Math::binomial_cdf_c ( unsigned int k, double p, unsigned int n )

Complement of the cumulative distribution function of the Binomial distribution.

Upper tail of the integral of the binomial_pdf

Definition at line 293 of file ProbFuncMathCore.cxx.

## ◆ breitwigner_cdf()

 double ROOT::Math::breitwigner_cdf ( double x, double gamma, double x0 = 0 )

Cumulative distribution function (lower tail) of the Breit_Wigner distribution and it is similar (just a different parameter definition) to the Cauchy distribution (see cauchy_cdf )

$D(x) = \int_{-\infty}^{x} \frac{1}{\pi} \frac{b}{x'^2 + (\frac{1}{2} \Gamma)^2} dx'$

Definition at line 39 of file ProbFuncMathCore.cxx.

## ◆ breitwigner_cdf_c()

 double ROOT::Math::breitwigner_cdf_c ( double x, double gamma, double x0 = 0 )

Complement of the cumulative distribution function (upper tail) of the Breit_Wigner distribution and it is similar (just a different parameter definition) to the Cauchy distribution (see cauchy_cdf_c )

$D(x) = \int_{x}^{+\infty} \frac{1}{\pi} \frac{\frac{1}{2} \Gamma}{x'^2 + (\frac{1}{2} \Gamma)^2} dx'$

Definition at line 33 of file ProbFuncMathCore.cxx.

## ◆ cauchy_cdf()

 double ROOT::Math::cauchy_cdf ( double x, double b, double x0 = 0 )

Cumulative distribution function (lower tail) of the Cauchy distribution which is also Lorentzian distribution.

It is similar (just a different parameter definition) to the Breit_Wigner distribution (see breitwigner_cdf )

$D(x) = \int_{-\infty}^{x} \frac{1}{\pi} \frac{ b }{ (x'-m)^2 + b^2} dx'$

For detailed description see Mathworld.

Definition at line 51 of file ProbFuncMathCore.cxx.

## ◆ cauchy_cdf_c()

 double ROOT::Math::cauchy_cdf_c ( double x, double b, double x0 = 0 )

Complement of the cumulative distribution function (upper tail) of the Cauchy distribution which is also Lorentzian distribution.

It is similar (just a different parameter definition) to the Breit_Wigner distribution (see breitwigner_cdf_c )

$D(x) = \int_{x}^{+\infty} \frac{1}{\pi} \frac{ b }{ (x'-m)^2 + b^2} dx'$

For detailed description see Mathworld.

Definition at line 45 of file ProbFuncMathCore.cxx.

## ◆ chisquared_cdf()

 double ROOT::Math::chisquared_cdf ( double x, double r, double x0 = 0 )

Cumulative distribution function of the $$\chi^2$$ distribution with $$r$$ degrees of freedom (lower tail).

$D_{r}(x) = \int_{-\infty}^{x} \frac{1}{\Gamma(r/2) 2^{r/2}} x'^{r/2-1} e^{-x'/2} dx'$

For detailed description see Mathworld. It is implemented using the incomplete gamma function, ROOT::Math::inc_gamma_c, from Cephes

Definition at line 63 of file ProbFuncMathCore.cxx.

## ◆ chisquared_cdf_c()

 double ROOT::Math::chisquared_cdf_c ( double x, double r, double x0 = 0 )

Complement of the cumulative distribution function of the $$\chi^2$$ distribution with $$r$$ degrees of freedom (upper tail).

$D_{r}(x) = \int_{x}^{+\infty} \frac{1}{\Gamma(r/2) 2^{r/2}} x'^{r/2-1} e^{-x'/2} dx'$

For detailed description see Mathworld. It is implemented using the incomplete gamma function, ROOT::Math::inc_gamma_c, from Cephes

Definition at line 57 of file ProbFuncMathCore.cxx.

## ◆ crystalball_cdf()

 double ROOT::Math::crystalball_cdf ( double x, double alpha, double n, double sigma, double x0 = 0 )

Cumulative distribution for the Crystal Ball distribution function.

See the definition of the Crystal Ball function at Wikipedia.

The distribution is defined only for n > 1 when the integral converges

Definition at line 69 of file ProbFuncMathCore.cxx.

## ◆ crystalball_cdf_c()

 double ROOT::Math::crystalball_cdf_c ( double x, double alpha, double n, double sigma, double x0 = 0 )

Complement of the Cumulative distribution for the Crystal Ball distribution.

See the definition of the Crystal Ball function at Wikipedia.

The distribution is defined only for n > 1 when the integral converges

Definition at line 84 of file ProbFuncMathCore.cxx.

## ◆ crystalball_integral()

 double ROOT::Math::crystalball_integral ( double x, double alpha, double n, double sigma, double x0 = 0 )

Integral of the not-normalized Crystal Ball function.

See the definition of the Crystal Ball function at Wikipedia.

see ROOT::Math::crystalball_function for the function evaluation.

Definition at line 98 of file ProbFuncMathCore.cxx.

## ◆ exponential_cdf()

 double ROOT::Math::exponential_cdf ( double x, double lambda, double x0 = 0 )

Cumulative distribution function of the exponential distribution (lower tail).

$D(x) = \int_{-\infty}^{x} \lambda e^{-\lambda x'} dx'$

For detailed description see Mathworld.

Definition at line 161 of file ProbFuncMathCore.cxx.

## ◆ exponential_cdf_c()

 double ROOT::Math::exponential_cdf_c ( double x, double lambda, double x0 = 0 )

Complement of the cumulative distribution function of the exponential distribution (upper tail).

$D(x) = \int_{x}^{+\infty} \lambda e^{-\lambda x'} dx'$

For detailed description see Mathworld.

Definition at line 154 of file ProbFuncMathCore.cxx.

## ◆ fdistribution_cdf()

 double ROOT::Math::fdistribution_cdf ( double x, double n, double m, double x0 = 0 )

Cumulative distribution function of the F-distribution (lower tail).

$D_{n,m}(x) = \int_{-\infty}^{x} \frac{\Gamma(\frac{n+m}{2})}{\Gamma(\frac{n}{2}) \Gamma(\frac{m}{2})} n^{n/2} m^{m/2} x'^{n/2 -1} (m+nx')^{-(n+m)/2} dx'$

For detailed description see Mathworld. It is implemented using the incomplete beta function, ROOT::Math::inc_beta, from Cephes

Definition at line 183 of file ProbFuncMathCore.cxx.

## ◆ fdistribution_cdf_c()

 double ROOT::Math::fdistribution_cdf_c ( double x, double n, double m, double x0 = 0 )

Complement of the cumulative distribution function of the F-distribution (upper tail).

$D_{n,m}(x) = \int_{x}^{+\infty} \frac{\Gamma(\frac{n+m}{2})}{\Gamma(\frac{n}{2}) \Gamma(\frac{m}{2})} n^{n/2} m^{m/2} x'^{n/2 -1} (m+nx')^{-(n+m)/2} dx'$

For detailed description see Mathworld. It is implemented using the incomplete beta function, ROOT::Math::inc_beta, from Cephes

Definition at line 169 of file ProbFuncMathCore.cxx.

## ◆ gamma_cdf()

 double ROOT::Math::gamma_cdf ( double x, double alpha, double theta, double x0 = 0 )

Cumulative distribution function of the gamma distribution (lower tail).

$D(x) = \int_{-\infty}^{x} {1 \over \Gamma(\alpha) \theta^{\alpha}} x'^{\alpha-1} e^{-x'/\theta} dx'$

For detailed description see Mathworld. It is implemented using the incomplete gamma function, ROOT::Math::inc_gamma, from Cephes

Definition at line 204 of file ProbFuncMathCore.cxx.

## ◆ gamma_cdf_c()

 double ROOT::Math::gamma_cdf_c ( double x, double alpha, double theta, double x0 = 0 )

Complement of the cumulative distribution function of the gamma distribution (upper tail).

$D(x) = \int_{x}^{+\infty} {1 \over \Gamma(\alpha) \theta^{\alpha}} x'^{\alpha-1} e^{-x'/\theta} dx'$

For detailed description see Mathworld. It is implemented using the incomplete gamma function, ROOT::Math::inc_gamma, from Cephes

Definition at line 198 of file ProbFuncMathCore.cxx.

## ◆ landau_cdf()

 double ROOT::Math::landau_cdf ( double x, double xi = 1, double x0 = 0 )

Cumulative distribution function of the Landau distribution (lower tail).

$D(x) = \int_{-\infty}^{x} p(x) dx$

where $$p(x)$$ is the Landau probability density function :

$p(x) = \frac{1}{\xi} \phi (\lambda)$

with

$\phi(\lambda) = \frac{1}{2 \pi i}\int_{c-i\infty}^{c+i\infty} e^{\lambda s + s \log{s}} ds$

with $$\lambda = (x-x_0)/\xi$$. 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 (DISLAN) is used.

Parameters
 x The argument $$x$$ xi The width parameter $$\xi$$ x0 The location parameter $$x_0$$

Definition at line 336 of file ProbFuncMathCore.cxx.

## ◆ landau_cdf_c()

 double ROOT::Math::landau_cdf_c ( double x, double xi = 1, double x0 = 0 )
inline

Complement of the distribution function of the Landau distribution (upper tail).

$D(x) = \int_{x}^{+\infty} p(x) dx$

where p(x) is the Landau probability density function. It is implemented simply as 1. - landau_cdf

Parameters
 x The argument $$x$$ xi The width parameter $$\xi$$ x0 The location parameter $$x_0$$

Definition at line 402 of file ProbFuncMathCore.h.

## ◆ lognormal_cdf()

 double ROOT::Math::lognormal_cdf ( double x, double m, double s, double x0 = 0 )

Cumulative distribution function of the lognormal distribution (lower tail).

$D(x) = \int_{-\infty}^{x} {1 \over x' \sqrt{2 \pi s^2} } e^{-(\ln{x'} - m)^2/2 s^2} dx'$

For detailed description see Mathworld.

Definition at line 218 of file ProbFuncMathCore.cxx.

## ◆ lognormal_cdf_c()

 double ROOT::Math::lognormal_cdf_c ( double x, double m, double s, double x0 = 0 )

Complement of the cumulative distribution function of the lognormal distribution (upper tail).

$D(x) = \int_{x}^{+\infty} {1 \over x' \sqrt{2 \pi s^2} } e^{-(\ln{x'} - m)^2/2 s^2} dx'$

For detailed description see Mathworld.

Definition at line 210 of file ProbFuncMathCore.cxx.

## ◆ negative_binomial_cdf()

 double ROOT::Math::negative_binomial_cdf ( unsigned int k, double p, double n )

Cumulative distribution function of the Negative Binomial distribution Lower tail of the integral of the negative_binomial_pdf.

Definition at line 316 of file ProbFuncMathCore.cxx.

## ◆ negative_binomial_cdf_c()

 double ROOT::Math::negative_binomial_cdf_c ( unsigned int k, double p, double n )

Complement of the cumulative distribution function of the Negative Binomial distribution.

Upper tail of the integral of the negative_binomial_pdf

Definition at line 326 of file ProbFuncMathCore.cxx.

## ◆ normal_cdf()

 double ROOT::Math::normal_cdf ( double x, double sigma = 1, double x0 = 0 )

Cumulative distribution function of the normal (Gaussian) distribution (lower tail).

$D(x) = \int_{-\infty}^{x} {1 \over \sqrt{2 \pi \sigma^2}} e^{-x'^2 / 2\sigma^2} dx'$

For detailed description see Mathworld.

Definition at line 234 of file ProbFuncMathCore.cxx.

## ◆ normal_cdf_c()

 double ROOT::Math::normal_cdf_c ( double x, double sigma = 1, double x0 = 0 )

Complement of the cumulative distribution function of the normal (Gaussian) distribution (upper tail).

$D(x) = \int_{x}^{+\infty} {1 \over \sqrt{2 \pi \sigma^2}} e^{-x'^2 / 2\sigma^2} dx'$

For detailed description see Mathworld.

Definition at line 226 of file ProbFuncMathCore.cxx.

## ◆ poisson_cdf()

 double ROOT::Math::poisson_cdf ( unsigned int n, double mu )

Cumulative distribution function of the Poisson distribution Lower tail of the integral of the poisson_pdf.

Definition at line 284 of file ProbFuncMathCore.cxx.

## ◆ poisson_cdf_c()

 double ROOT::Math::poisson_cdf_c ( unsigned int n, double mu )

Complement of the cumulative distribution function of the Poisson distribution.

discrete distributions

Upper tail of the integral of the poisson_pdf

Definition at line 275 of file ProbFuncMathCore.cxx.

## ◆ tdistribution_cdf()

 double ROOT::Math::tdistribution_cdf ( double x, double r, double x0 = 0 )

Cumulative distribution function of Student's t-distribution (lower tail).

$D_{r}(x) = \int_{-\infty}^{x} \frac{\Gamma(\frac{r+1}{2})}{\sqrt{r \pi}\Gamma(\frac{r}{2})} \left( 1+\frac{x'^2}{r}\right)^{-(r+1)/2} dx'$

For detailed description see Mathworld. It is implemented using the incomplete beta function, ROOT::Math::inc_beta, from Cephes

Definition at line 250 of file ProbFuncMathCore.cxx.

## ◆ tdistribution_cdf_c()

 double ROOT::Math::tdistribution_cdf_c ( double x, double r, double x0 = 0 )

Complement of the cumulative distribution function of Student's t-distribution (upper tail).

$D_{r}(x) = \int_{x}^{+\infty} \frac{\Gamma(\frac{r+1}{2})}{\sqrt{r \pi}\Gamma(\frac{r}{2})} \left( 1+\frac{x'^2}{r}\right)^{-(r+1)/2} dx'$

For detailed description see Mathworld. It is implemented using the incomplete beta function, ROOT::Math::inc_beta, from Cephes

Definition at line 242 of file ProbFuncMathCore.cxx.

## ◆ uniform_cdf()

 double ROOT::Math::uniform_cdf ( double x, double a, double b, double x0 = 0 )

Cumulative distribution function of the uniform (flat) distribution (lower tail).

$D(x) = \int_{-\infty}^{x} {1 \over (b-a)} dx'$

For detailed description see Mathworld.

Definition at line 266 of file ProbFuncMathCore.cxx.

## ◆ uniform_cdf_c()

 double ROOT::Math::uniform_cdf_c ( double x, double a, double b, double x0 = 0 )

Complement of the cumulative distribution function of the uniform (flat) distribution (upper tail).

$D(x) = \int_{x}^{+\infty} {1 \over (b-a)} dx'$

For detailed description see Mathworld.

Definition at line 258 of file ProbFuncMathCore.cxx.

## ◆ vavilov_accurate_cdf()

 double ROOT::Math::vavilov_accurate_cdf ( double x, double kappa, double beta2 )

The Vavilov cumulative probability density function.

Parameters
 x The Landau parameter $$x = \lambda_L$$ kappa The parameter $$\kappa$$, which must be in the range $$\kappa \ge 0.001$$ beta2 The parameter $$\beta^2$$, which must be in the range $$0 \le \beta^2 \le 1$$

Definition at line 471 of file VavilovAccurate.cxx.

## ◆ vavilov_accurate_cdf_c()

 double ROOT::Math::vavilov_accurate_cdf_c ( double x, double kappa, double beta2 )

The Vavilov complementary cumulative probability density function.

Parameters
 x The Landau parameter $$x = \lambda_L$$ kappa The parameter $$\kappa$$, which must be in the range $$\kappa \ge 0.001$$ beta2 The parameter $$\beta^2$$, which must be in the range $$0 \le \beta^2 \le 1$$

Definition at line 466 of file VavilovAccurate.cxx.

## ◆ vavilov_fast_cdf()

 double ROOT::Math::vavilov_fast_cdf ( double x, double kappa, double beta2 )

The Vavilov cumulative probability density function.

Parameters
 x The Landau parameter $$x = \lambda_L$$ kappa The parameter $$\kappa$$, which must be in the range $$0.01 \le \kappa \le 12$$ beta2 The parameter $$\beta^2$$, which must be in the range $$0 \le \beta^2 \le 1$$

Definition at line 582 of file VavilovFast.cxx.

## ◆ vavilov_fast_cdf_c()

 double ROOT::Math::vavilov_fast_cdf_c ( double x, double kappa, double beta2 )

The Vavilov complementary cumulative probability density function.

Parameters
 x The Landau parameter $$x = \lambda_L$$ kappa The parameter $$\kappa$$, which must be in the range $$0.01 \le \kappa \le 12$$ beta2 The parameter $$\beta^2$$, which must be in the range $$0 \le \beta^2 \le 1$$

Definition at line 587 of file VavilovFast.cxx.