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ROOT
Reference Guide |
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ROOT
Reference Guide |
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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. More... | |
| double | ROOT::Math::beta_cdf_c (double x, double a, double b) |
| Complement of the cumulative distribution function of the beta distribution. More... | |
| 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. More... | |
| double | ROOT::Math::binomial_cdf_c (unsigned int k, double p, unsigned int n) |
| Complement of the cumulative distribution function of the Binomial distribution. More... | |
| 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 ) More... | |
| 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 ) More... | |
| 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. More... | |
| 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. More... | |
| 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). More... | |
| 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). More... | |
| double | ROOT::Math::crystalball_cdf (double x, double alpha, double n, double sigma, double x0=0) |
| Cumulative distribution for the Crystal Ball distribution function. More... | |
| 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. More... | |
| double | ROOT::Math::crystalball_integral (double x, double alpha, double n, double sigma, double x0=0) |
| Integral of the not-normalized Crystal Ball function. More... | |
| double | ROOT::Math::exponential_cdf (double x, double lambda, double x0=0) |
| Cumulative distribution function of the exponential distribution (lower tail). More... | |
| 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). More... | |
| double | ROOT::Math::fdistribution_cdf (double x, double n, double m, double x0=0) |
| Cumulative distribution function of the F-distribution (lower tail). More... | |
| 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). More... | |
| double | ROOT::Math::gamma_cdf (double x, double alpha, double theta, double x0=0) |
| Cumulative distribution function of the gamma distribution (lower tail). More... | |
| 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). More... | |
| double | ROOT::Math::landau_cdf (double x, double xi=1, double x0=0) |
| Cumulative distribution function of the Landau distribution (lower tail). More... | |
| 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). More... | |
| double | ROOT::Math::lognormal_cdf (double x, double m, double s, double x0=0) |
| Cumulative distribution function of the lognormal distribution (lower tail). More... | |
| 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). More... | |
| 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. More... | |
| 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. More... | |
| double | ROOT::Math::normal_cdf (double x, double sigma=1, double x0=0) |
| Cumulative distribution function of the normal (Gaussian) distribution (lower tail). More... | |
| 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). More... | |
| 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. More... | |
| double | ROOT::Math::poisson_cdf_c (unsigned int n, double mu) |
| Complement of the cumulative distribution function of the Poisson distribution. More... | |
| double | ROOT::Math::tdistribution_cdf (double x, double r, double x0=0) |
| Cumulative distribution function of Student's t-distribution (lower tail). More... | |
| 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). More... | |
| double | ROOT::Math::uniform_cdf (double x, double a, double b, double x0=0) |
| Cumulative distribution function of the uniform (flat) distribution (lower tail). More... | |
| 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). More... | |
| double | ROOT::Math::vavilov_accurate_cdf (double x, double kappa, double beta2) |
| The Vavilov cumulative probability density function. More... | |
| double | ROOT::Math::vavilov_accurate_cdf_c (double x, double kappa, double beta2) |
| The Vavilov complementary cumulative probability density function. More... | |
| double | ROOT::Math::vavilov_fast_cdf (double x, double kappa, double beta2) |
| The Vavilov cumulative probability density function. More... | |
| double | ROOT::Math::vavilov_fast_cdf_c (double x, double kappa, double beta2) |
| The Vavilov complementary cumulative probability density function. More... | |
Cumulative distribution function of the beta distribution Upper tail of the integral of the beta_pdf.
Definition at line 27 of file ProbFuncMathCore.cxx.
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.
Cumulative distribution function of the Binomial distribution Lower tail of the integral of the binomial_pdf.
Definition at line 304 of file ProbFuncMathCore.cxx.
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.
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.
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.
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.
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.
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.
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.
| 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.
| 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.
| 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.
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.
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.
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.
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.
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.
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.
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.
| x | The argument \(x\) |
| xi | The width parameter \(\xi\) |
| x0 | The location parameter \(x_0\) |
Definition at line 336 of file ProbFuncMathCore.cxx.
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
| x | The argument \(x\) |
| xi | The width parameter \(\xi\) |
| x0 | The location parameter \(x_0\) |
Definition at line 402 of file ProbFuncMathCore.h.
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.
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.
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.
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.
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.
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.
Cumulative distribution function of the Poisson distribution Lower tail of the integral of the poisson_pdf.
Definition at line 284 of file ProbFuncMathCore.cxx.
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.
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.
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.
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.
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.
The Vavilov cumulative probability density function.
| 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 472 of file VavilovAccurate.cxx.
The Vavilov complementary cumulative probability density function.
| 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 467 of file VavilovAccurate.cxx.
The Vavilov cumulative probability density function.
| 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.
The Vavilov complementary cumulative probability density function.
| 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.
