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ROOT
Reference Guide |
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ROOT
Reference Guide |
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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.
Functions | |
| double | ROOT::Math::noncentral_chisquared_pdf (double x, double r, double lambda) |
| Probability density function of the non central \(\chi^2\) distribution with \(r\) degrees of freedom and the noon-central parameter \(\lambda\). | |
| double | ROOT::Math::vavilov_accurate_pdf (double x, double kappa, double beta2) |
| The Vavilov probability density function. | |
| double | ROOT::Math::vavilov_fast_pdf (double x, double kappa, double beta2) |
| The Vavilov probability density function. | |
Probability Density Functions from MathCore | |
Additional PDFs are provided in the MathMore library when the GSL is available. | |
| double | ROOT::Math::beta_pdf (double x, double a, double b) |
| Probability density function of the beta distribution. | |
| double | ROOT::Math::binomial_pdf (unsigned int k, double p, unsigned int n) |
| Probability density function of the binomial distribution. | |
| double | ROOT::Math::negative_binomial_pdf (unsigned int k, double p, double n) |
| Probability density function of the negative binomial distribution. | |
| double | ROOT::Math::breitwigner_pdf (double x, double gamma, double x0=0) |
| Probability density function of Breit-Wigner distribution, which is similar, just a different definition of the parameters, to the Cauchy distribution (see cauchy_pdf ) | |
| double | ROOT::Math::cauchy_pdf (double x, double b=1, double x0=0) |
| Probability density function of the Cauchy distribution which is also called Lorentzian distribution. | |
| double | ROOT::Math::chisquared_pdf (double x, double r, double x0=0) |
| Probability density function of the \(\chi^2\) distribution with \(r\) degrees of freedom. | |
| double | ROOT::Math::crystalball_function (double x, double alpha, double n, double sigma, double mean=0) |
| Crystal ball function. | |
| double | ROOT::Math::exponential_pdf (double x, double lambda, double x0=0) |
| Probability density function of the exponential distribution. | |
| double | ROOT::Math::fdistribution_pdf (double x, double n, double m, double x0=0) |
| Probability density function of the F-distribution. | |
| double | ROOT::Math::gamma_pdf (double x, double alpha, double theta, double x0=0) |
| Probability density function of the gamma distribution. | |
| double | ROOT::Math::gaussian_pdf (double x, double sigma=1, double x0=0) |
| Probability density function of the normal (Gaussian) distribution with mean x0 and standard deviation sigma. | |
| double | ROOT::Math::bigaussian_pdf (double x, double y, double sigmax=1, double sigmay=1, double rho=0, double x0=0, double y0=0) |
| Probability density function of the bi-dimensional (Gaussian) distribution. | |
| double | ROOT::Math::landau_pdf (double x, double xi=1, double x0=0) |
| Probability density function of the Landau distribution: | |
| double | ROOT::Math::lognormal_pdf (double x, double m, double s, double x0=0) |
| Probability density function of the lognormal distribution. | |
| double | ROOT::Math::normal_pdf (double x, double sigma=1, double x0=0) |
| Probability density function of the normal (Gaussian) distribution with mean x0 and standard deviation sigma. | |
| double | ROOT::Math::poisson_pdf (unsigned int n, double mu) |
| Probability density function of the Poisson distribution. | |
| double | ROOT::Math::tdistribution_pdf (double x, double r, double x0=0) |
| Probability density function of Student's t-distribution. | |
| double | ROOT::Math::uniform_pdf (double x, double a, double b, double x0=0) |
| Probability density function of the uniform (flat) distribution. | |
| double | ROOT::Math::crystalball_pdf (double x, double alpha, double n, double sigma, double mean=0) |
| pdf definition of the crystal_ball which is defined only for n > 1 otherwise integral is diverging | |
Probability density function of the beta distribution.
\[ p(x) = \frac{\Gamma (a + b) } {\Gamma(a)\Gamma(b) } x ^{a-1} (1 - x)^{b-1} \]
for \(0 \leq x \leq 1 \). For detailed description see Mathworld.
Definition at line 75 of file PdfFuncMathCore.h.
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Probability density function of the bi-dimensional (Gaussian) distribution.
\[ p(x,y) = {1 \over 2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp (-((x-x0)^2/\sigma_x^2 + (y-y0)^2/\sigma_y^2 - 2 \rho x y/(\sigma_x\sigma_y))/2(1-\rho^2)) \]
For detailed description see Mathworld. It can also be evaluated using normal_pdf which will call the same implementation.
| x | x variable |
| y | y variable |
| sigmax | the stdev in x |
| sigmay | the stdev in y |
| rho | correlation, must be between -1,1 |
| x0 | the offset in x |
| y0 | the offset in y |
Definition at line 424 of file PdfFuncMathCore.h.
Probability density function of the binomial distribution.
\[ p(k) = \frac{n!}{k! (n-k)!} p^k (1-p)^{n-k} \]
for \( 0 \leq k \leq n \). For detailed description see Mathworld.
Definition at line 111 of file PdfFuncMathCore.h.
Probability density function of Breit-Wigner distribution, which is similar, just a different definition of the parameters, to the Cauchy distribution (see cauchy_pdf )
\[ p(x) = \frac{1}{\pi} \frac{\frac{1}{2} \Gamma}{x^2 + (\frac{1}{2} \Gamma)^2} \]
Definition at line 168 of file PdfFuncMathCore.h.
Probability density function of the Cauchy distribution which is also called Lorentzian distribution.
\[ p(x) = \frac{1}{\pi} \frac{ b }{ (x-m)^2 + b^2} \]
For detailed description see Mathworld. It is also related to the breitwigner_pdf which will call the same implementation.
Definition at line 194 of file PdfFuncMathCore.h.
Probability density function of the \(\chi^2\) distribution with \(r\) degrees of freedom.
\[ p_r(x) = \frac{1}{\Gamma(r/2) 2^{r/2}} x^{r/2-1} e^{-x/2} \]
for \(x \geq 0\). For detailed description see Mathworld.
Definition at line 218 of file PdfFuncMathCore.h.
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Crystal ball function.
See the definition at Wikipedia.
It is not really a pdf since it is not normalized
Definition at line 247 of file PdfFuncMathCore.h.
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pdf definition of the crystal_ball which is defined only for n > 1 otherwise integral is diverging
Definition at line 271 of file PdfFuncMathCore.h.
Probability density function of the exponential distribution.
\[ p(x) = \lambda e^{-\lambda x} \]
for x>0. For detailed description see Mathworld.
Definition at line 299 of file PdfFuncMathCore.h.
Probability density function of the F-distribution.
\[ p_{n,m}(x) = \frac{\Gamma(\frac{n+m}{2})}{\Gamma(\frac{n}{2}) \Gamma(\frac{m}{2})} n^{n/2} m^{m/2} x^{n/2 -1} (m+nx)^{-(n+m)/2} \]
for x>=0. For detailed description see Mathworld.
Definition at line 325 of file PdfFuncMathCore.h.
Probability density function of the gamma distribution.
\[ p(x) = {1 \over \Gamma(\alpha) \theta^{\alpha}} x^{\alpha-1} e^{-x/\theta} \]
for x>0. For detailed description see Mathworld.
Definition at line 356 of file PdfFuncMathCore.h.
Probability density function of the normal (Gaussian) distribution with mean x0 and standard deviation sigma.
\[ p(x) = {1 \over \sqrt{2 \pi \sigma^2}} e^{-(x-x_0)^2 / 2\sigma^2} \]
For detailed description see Mathworld. It can also be evaluated using normal_pdf which will call the same implementation.
Definition at line 395 of file PdfFuncMathCore.h.
Probability density function of the Landau distribution:
\[ p(x) = \frac{1}{\xi} \phi (\lambda) \]
with
\[ \phi(\lambda) = \frac{1}{2 \pi i}\int_{c-i\infty}^{c+i\infty} e^{\lambda s + s \log{s}} ds\]
where \(\lambda = (x-x_0)/\xi\). 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 (DENLAN) is used
| x | The argument \(x\) |
| xi | The width parameter \(\xi\) |
| x0 | The location parameter \(x_0\) |
Definition at line 21 of file PdfFuncMathCore.cxx.
Probability density function of the lognormal distribution.
\[ p(x) = {1 \over x \sqrt{2 \pi s^2} } e^{-(\ln{x} - m)^2/2 s^2} \]
for x>0. For detailed description see Mathworld.
| x | x variable |
| m | M = 0 for lognormal |
| s | scale parameter (not the sigma of the distribution which is not even defined) |
| x0 | location parameter, corresponds approximately to the most probable value. For x0 = 0, sigma = 1, the x_mpv = -0.22278 |
Definition at line 476 of file PdfFuncMathCore.h.
Probability density function of the negative binomial distribution.
\[ p(k) = \frac{(k+n-1)!}{k! (n-1)!} p^{n} (1-p)^{k} \]
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.
Definition at line 139 of file PdfFuncMathCore.h.
Probability density function of the non central \(\chi^2\) distribution with \(r\) degrees of freedom and the noon-central parameter \(\lambda\).
\[ p_r(x) = \frac{1}{\Gamma(r/2) 2^{r/2}} x^{r/2-1} e^{-x/2} \]
for \(x \geq 0\). For detailed description see Mathworld.
Definition at line 22 of file PdfFuncMathMore.cxx.
Probability density function of the normal (Gaussian) distribution with mean x0 and standard deviation sigma.
\[ p(x) = {1 \over \sqrt{2 \pi \sigma^2}} e^{-(x-x_0)^2 / 2\sigma^2} \]
For detailed description see Mathworld. It can also be evaluated using gaussian_pdf which will call the same implementation.
Definition at line 502 of file PdfFuncMathCore.h.
Probability density function of the Poisson distribution.
\[ p(n) = \frac{\mu^n}{n!} e^{- \mu} \]
For detailed description see Mathworld.
Definition at line 525 of file PdfFuncMathCore.h.
Probability density function of Student's t-distribution.
\[ p_{r}(x) = \frac{\Gamma(\frac{r+1}{2})}{\sqrt{r \pi}\Gamma(\frac{r}{2})} \left( 1+\frac{x^2}{r}\right)^{-(r+1)/2} \]
for \(r \geq 0\). For detailed description see Mathworld.
Definition at line 556 of file PdfFuncMathCore.h.
Probability density function of the uniform (flat) distribution.
\[ p(x) = {1 \over (b-a)} \]
if \(a \leq x<b\) and 0 otherwise. For detailed description see Mathworld.
Definition at line 581 of file PdfFuncMathCore.h.
The Vavilov 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 461 of file VavilovAccurate.cxx.
The Vavilov 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 577 of file VavilovFast.cxx.
