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Probability Density Functions (PDF) |
| 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.
Additional PDF's are provided in the MathMore library.
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| double | ROOT::Math::binomial_pdf (unsigned int k, double p, unsigned int n) |
| | Probability density function of the binomial distribution.
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| 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 ).
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| 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.
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| double | ROOT::Math::chisquared_pdf (double x, double r, double x0=0) |
| | Probability density function of the  distribution with  degrees of freedom.
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| double | ROOT::Math::exponential_pdf (double x, double lambda, double x0=0) |
| | Probability density function of the exponential distribution.
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| double | ROOT::Math::fdistribution_pdf (double x, double n, double m, double x0=0) |
| | Probability density function of the F-distribution.
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| double | ROOT::Math::gamma_pdf (double x, double alpha, double theta, double x0=0) |
| | Probability density function of the gamma distribution.
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| double | ROOT::Math::gaussian_pdf (double x, double sigma, double x0=0) |
| | Probability density function of the normal (Gaussian) distribution.
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| double | ROOT::Math::lognormal_pdf (double x, double m, double s, double x0=0) |
| | double landau_pdf(double x) {
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| double | ROOT::Math::normal_pdf (double x, double sigma, double x0=0) |
| | Probability density function of the normal (Gaussian) distribution.
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| double | ROOT::Math::poisson_pdf (unsigned int n, double mu) |
| | Probability density function of the Poisson distribution.
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| double | ROOT::Math::tdistribution_pdf (double x, double r, double x0=0) |
| | Probability density function of Student's t-distribution.
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| double | ROOT::Math::uniform_pdf (double x, double a, double b, double x0=0) |
| | Probability density function of the uniform (flat) distribution.
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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
![\[ D(x) = \int_{-\infty}^{x} p(x') dx' \]](form_18.png)
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
![\[ D(x) = \int_{x}^{+\infty} p(x') dx' \]](form_19.png)
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.
Additional CDF's are provided in the MathMore library.
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| 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 ).
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| 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 ).
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| 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.
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| 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.
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| double | ROOT::Math::exponential_cdf_c (double x, double lambda, double x0=0) |
| | double chisquared_cdf_c(double x, double r) {
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| double | ROOT::Math::exponential_cdf (double x, double lambda, double x0=0) |
| | Cumulative distribution function of the exponential distribution (lower tail).
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| double | ROOT::Math::gaussian_cdf_c (double x, double sigma, double x0=0) |
| | double fdistribution_cdf_c(double x, double n, double m) {
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| double | ROOT::Math::gaussian_cdf (double x, double sigma, double x0=0) |
| | Cumulative distribution function of the normal (Gaussian) distribution (lower tail).
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| 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).
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| double | ROOT::Math::lognormal_cdf (double x, double m, double s, double x0=0) |
| | Cumulative distribution function of the lognormal distribution (lower tail).
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| double | ROOT::Math::normal_cdf_c (double x, double sigma, double x0=0) |
| | Complement of the cumulative distribution function of the normal (Gaussian) distribution (upper tail).
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| double | ROOT::Math::normal_cdf (double x, double sigma, double x0=0) |
| | Cumulative distribution function of the normal (Gaussian) distribution (lower tail).
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| double | ROOT::Math::uniform_cdf_c (double x, double a, double b, double x0=0) |
| | double tdistribution_cdf_c(double x, double r) {
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| double | ROOT::Math::uniform_cdf (double x, double a, double b, double x0=0) |
| | Cumulative distribution function of the uniform (flat) distribution (lower tail).
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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 funcitons and a sub-set of the cumulative distributions. Additional distributions and all the inverses cumulative distributions (quantiles) are provided by MathMore library.
![\[ p(k) = \frac{n!}{k! (n-k)!} p^k (1-p)^{n-k} \]](form_0.png)
. For detailed description see ![\[ D(x) = \int_{-\infty}^{x} \frac{1}{\pi} \frac{b}{x'^2 + (\frac{1}{2} \Gamma)^2} dx' \]](form_21.png)
![\[ D(x) = \int_{x}^{+\infty} \frac{1}{\pi} \frac{\frac{1}{2} \Gamma}{x'^2 + (\frac{1}{2} \Gamma)^2} dx' \]](form_20.png)
![\[ p(x) = \frac{1}{\pi} \frac{\frac{1}{2} \Gamma}{x^2 + (\frac{1}{2} \Gamma)^2} \]](form_2.png)
![\[ D(x) = \int_{-\infty}^{x} \frac{1}{\pi} \frac{ b }{ (x'-m)^2 + b^2} dx' \]](form_23.png)
![\[ D(x) = \int_{x}^{+\infty} \frac{1}{\pi} \frac{ b }{ (x'-m)^2 + b^2} dx' \]](form_22.png)
![\[ p(x) = \frac{1}{\pi} \frac{ b }{ (x-m)^2 + b^2} \]](form_3.png)
![\[ p_r(x) = \frac{1}{\Gamma(r/2) 2^{r/2}} x^{r/2-1} e^{-x/2} \]](form_6.png)
. For detailed description see ![\[ D(x) = \int_{-\infty}^{x} \lambda e^{-\lambda x'} dx' \]](form_25.png)
![\[ D(x) = \int_{x}^{+\infty} \lambda e^{-\lambda x'} dx' \]](form_24.png)
![\[ p(x) = \lambda e^{-\lambda x} \]](form_8.png)
![\[ p_{n,m}(x) = \frac{\Gamma(\frac{n+m}{2})}{\Gamma(\frac{n}{2}) \Gamma(\frac{m}{2})} n^{n/2} m^{m/2} x^{n/2 -1} (m+nx)^{-(n+m)/2} \]](form_9.png)
![\[ p(x) = {1 \over \Gamma(\alpha) \theta^{\alpha}} x^{\alpha-1} e^{-x/\theta} \]](form_10.png)
![\[ D(x) = \int_{-\infty}^{x} {1 \over \sqrt{2 \pi \sigma^2}} e^{-x'^2 / 2\sigma^2} dx' \]](form_27.png)
![\[ D(x) = \int_{x}^{+\infty} {1 \over \sqrt{2 \pi \sigma^2}} e^{-x'^2 / 2\sigma^2} dx' \]](form_26.png)
![\[ p(x) = {1 \over \sqrt{2 \pi \sigma^2}} e^{-x^2 / 2\sigma^2} \]](form_11.png)
![\[ D(x) = \int_{-\infty}^{x} {1 \over x' \sqrt{2 \pi s^2} } e^{-(\ln{x'} - m)^2/2 s^2} dx' \]](form_29.png)
![\[ D(x) = \int_{x}^{+\infty} {1 \over x' \sqrt{2 \pi s^2} } e^{-(\ln{x'} - m)^2/2 s^2} dx' \]](form_28.png)
![\[ p(x) = {1 \over x \sqrt{2 \pi s^2} } e^{-(\ln{x} - m)^2/2 s^2} \]](form_12.png)
![\[ p(n) = \frac{\mu^n}{n!} e^{- \mu} \]](form_13.png)
![\[ p_{r}(x) = \frac{\Gamma(\frac{r+1}{2})}{\sqrt{r \pi}\Gamma(\frac{r}{2})} \left( 1+\frac{x^2}{r}\right)^{-(r+1)/2} \]](form_14.png)
. For detailed description see ![\[ D(x) = \int_{-\infty}^{x} {1 \over (b-a)} dx' \]](form_31.png)
![\[ D(x) = \int_{x}^{+\infty} {1 \over (b-a)} dx' \]](form_30.png)
![\[ p(x) = {1 \over (b-a)} \]](form_16.png)
and 0 otherwise. For detailed description see 