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PdfFuncMathCore.h
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1 // @(#)root/mathcore:$Id$
2 // Authors: Andras Zsenei & Lorenzo Moneta 06/2005
3 
4 /**********************************************************************
5  * *
6  * Copyright (c) 2005 , LCG ROOT MathLib Team *
7  * *
8  * *
9  **********************************************************************/
10 
11 
12 
13 /**
14 
15 Probability density functions, cumulative distribution functions
16 and their inverses (quantiles) for various statistical distributions (continuous and discrete).
17 Whenever possible the conventions followed are those of the
18 CRC Concise Encyclopedia of Mathematics, Second Edition
19 (or <A HREF="http://mathworld.wolfram.com/">Mathworld</A>).
20 By convention the distributions are centered around 0, so for
21 example in the case of a Gaussian there is no parameter mu. The
22 user must calculate the shift themselves if they wish.
23 
24 MathCore provides the majority of the probability density functions, of the
25 cumulative distributions and of the quantiles (inverses of the cumulatives).
26 Additional distributions are also provided by the
27 <A HREF="../../MathMore/html/group__StatFunc.html">MathMore</A> library.
28 
29 
30 @defgroup StatFunc Statistical functions
31 
32 @ingroup MathCore
33 @ingroup MathMore
34 
35 */
36 
37 #ifndef ROOT_Math_PdfFuncMathCore
38 #define ROOT_Math_PdfFuncMathCore
39 
40 #include "Math/Math.h"
41 #include "Math/SpecFuncMathCore.h"
42 
43 #include <limits>
44 
45 namespace ROOT {
46 namespace Math {
47 
48 
49 
50  /** @defgroup PdfFunc Probability Density Functions (PDF)
51  * @ingroup StatFunc
52  * Probability density functions of various statistical distributions
53  * (continuous and discrete).
54  * The probability density function returns the probability that
55  * the variate has the value x.
56  * In statistics the PDF is also called the frequency function.
57  *
58  *
59  */
60 
61  /** @name Probability Density Functions from MathCore
62  * Additional PDF's are provided in the MathMore library
63  * (see PDF functions from MathMore)
64  */
65 
66  //@{
67 
68  /**
69 
70  Probability density function of the beta distribution.
71 
72  \f[ p(x) = \frac{\Gamma (a + b) } {\Gamma(a)\Gamma(b) } x ^{a-1} (1 - x)^{b-1} \f]
73 
74  for \f$0 \leq x \leq 1 \f$. For detailed description see
75  <A HREF="http://mathworld.wolfram.com/BetaDistribution.html">
76  Mathworld</A>.
77 
78  @ingroup PdfFunc
79 
80  */
81 
82  inline double beta_pdf(double x, double a, double b) {
83  // Inlined to enable clad-auto-derivation for this function.
84 
85  if (x < 0 || x > 1.0) return 0;
86  if (x == 0 ) {
87  // need this work Windows
88  if (a < 1) return std::numeric_limits<double>::infinity();
89  else if (a > 1) return 0;
90  else if ( a == 1) return b; // to avoid a nan from log(0)*0
91  }
92  if (x == 1 ) {
93  // need this work Windows
94  if (b < 1) return std::numeric_limits<double>::infinity();
95  else if (b > 1) return 0;
96  else if ( b == 1) return a; // to avoid a nan from log(0)*0
97  }
99  std::log(x) * (a -1.) + ROOT::Math::log1p(-x ) * (b - 1.) );
100  }
101 
102 
103 
104  /**
105 
106  Probability density function of the binomial distribution.
107 
108  \f[ p(k) = \frac{n!}{k! (n-k)!} p^k (1-p)^{n-k} \f]
109 
110  for \f$ 0 \leq k \leq n \f$. For detailed description see
111  <A HREF="http://mathworld.wolfram.com/BinomialDistribution.html">
112  Mathworld</A>.
113 
114  @ingroup PdfFunc
115 
116  */
117 
118  inline double binomial_pdf(unsigned int k, double p, unsigned int n) {
119  // Inlined to enable clad-auto-derivation for this function.
120  if (k > n)
121  return 0.0;
122 
123  double coeff = ROOT::Math::lgamma(n+1) - ROOT::Math::lgamma(k+1) - ROOT::Math::lgamma(n-k+1);
124  return std::exp(coeff + k * std::log(p) + (n - k) * ROOT::Math::log1p(-p));
125  }
126 
127 
128 
129  /**
130 
131  Probability density function of the negative binomial distribution.
132 
133  \f[ p(k) = \frac{(k+n-1)!}{k! (n-1)!} p^{n} (1-p)^{k} \f]
134 
135  For detailed description see
136  <A HREF="http://mathworld.wolfram.com/NegativeBinomialDistribution.html">
137  Mathworld</A> (where \f$k \to x\f$ and \f$n \to r\f$).
138  The distribution in <A HREF="http://en.wikipedia.org/wiki/Negative_binomial_distribution">
139  Wikipedia</A> is defined with a \f$p\f$ corresponding to \f$1-p\f$ in this case.
140 
141 
142  @ingroup PdfFunc
143 
144  */
145 
146  inline double negative_binomial_pdf(unsigned int k, double p, double n) {
147  // Inlined to enable clad-auto-derivation for this function.
148 
149  // implement in term of gamma function
150 
151  if (n < 0) return 0.0;
152  if (p < 0 || p > 1.0) return 0.0;
153 
154  double coeff = ROOT::Math::lgamma(k+n) - ROOT::Math::lgamma(k+1.0) - ROOT::Math::lgamma(n);
155  return std::exp(coeff + n * std::log(p) + double(k) * ROOT::Math::log1p(-p));
156 
157  }
158 
159 
160 
161 
162  /**
163 
164  Probability density function of Breit-Wigner distribution, which is similar, just
165  a different definition of the parameters, to the Cauchy distribution
166  (see #cauchy_pdf )
167 
168  \f[ p(x) = \frac{1}{\pi} \frac{\frac{1}{2} \Gamma}{x^2 + (\frac{1}{2} \Gamma)^2} \f]
169 
170 
171  @ingroup PdfFunc
172 
173  */
174 
175  inline double breitwigner_pdf(double x, double gamma, double x0 = 0) {
176  // Inlined to enable clad-auto-derivation for this function.
177  double gammahalf = gamma/2.0;
178  return gammahalf/(M_PI * ((x-x0)*(x-x0) + gammahalf*gammahalf));
179  }
180 
181 
182 
183 
184  /**
185 
186  Probability density function of the Cauchy distribution which is also
187  called Lorentzian distribution.
188 
189 
190  \f[ p(x) = \frac{1}{\pi} \frac{ b }{ (x-m)^2 + b^2} \f]
191 
192  For detailed description see
193  <A HREF="http://mathworld.wolfram.com/CauchyDistribution.html">
194  Mathworld</A>. It is also related to the #breitwigner_pdf which
195  will call the same implementation.
196 
197  @ingroup PdfFunc
198 
199  */
200 
201  inline double cauchy_pdf(double x, double b = 1, double x0 = 0) {
202 
203  return b/(M_PI * ((x-x0)*(x-x0) + b*b));
204 
205  }
206 
207 
208 
209 
210  /**
211 
212  Probability density function of the \f$\chi^2\f$ distribution with \f$r\f$
213  degrees of freedom.
214 
215  \f[ p_r(x) = \frac{1}{\Gamma(r/2) 2^{r/2}} x^{r/2-1} e^{-x/2} \f]
216 
217  for \f$x \geq 0\f$. For detailed description see
218  <A HREF="http://mathworld.wolfram.com/Chi-SquaredDistribution.html">
219  Mathworld</A>.
220 
221  @ingroup PdfFunc
222 
223  */
224 
225  inline double chisquared_pdf(double x, double r, double x0 = 0) {
226  // Inlined to enable clad-auto-derivation for this function.
227 
228  if ((x-x0) < 0) {
229  return 0.0;
230  }
231  double a = r/2 -1.;
232  // let return inf for case x = x0 and treat special case of r = 2 otherwise will return nan
233  if (x == x0 && a == 0) return 0.5;
234 
235  return std::exp ((r/2 - 1) * std::log((x-x0)/2) - (x-x0)/2 - ROOT::Math::lgamma(r/2))/2;
236 
237  }
238 
239 
240  /**
241 
242  Crystal ball function
243 
244  See the definition at
245  <A HREF="http://en.wikipedia.org/wiki/Crystal_Ball_function">
246  Wikipedia</A>.
247 
248  It is not really a pdf since it is not normalized
249 
250  @ingroup PdfFunc
251 
252  */
253 
254  inline double crystalball_function(double x, double alpha, double n, double sigma, double mean = 0) {
255  // Inlined to enable clad-auto-derivation for this function.
256 
257  // evaluate the crystal ball function
258  if (sigma < 0.) return 0.;
259  double z = (x - mean)/sigma;
260  if (alpha < 0) z = -z;
261  double abs_alpha = std::abs(alpha);
262  // double C = n/abs_alpha * 1./(n-1.) * std::exp(-alpha*alpha/2.);
263  // double D = std::sqrt(M_PI/2.)*(1.+ROOT::Math::erf(abs_alpha/std::sqrt(2.)));
264  // double N = 1./(sigma*(C+D));
265  if (z > - abs_alpha)
266  return std::exp(- 0.5 * z * z);
267  //double A = std::pow(n/abs_alpha,n) * std::exp(-0.5*abs_alpha*abs_alpha);
268  double nDivAlpha = n/abs_alpha;
269  double AA = std::exp(-0.5*abs_alpha*abs_alpha);
270  double B = nDivAlpha -abs_alpha;
271  double arg = nDivAlpha/(B-z);
272  return AA * std::pow(arg,n);
273  }
274 
275  /**
276  pdf definition of the crystal_ball which is defined only for n > 1 otherwise integral is diverging
277  */
278  inline double crystalball_pdf(double x, double alpha, double n, double sigma, double mean = 0) {
279  // Inlined to enable clad-auto-derivation for this function.
280 
281  // evaluation of the PDF ( is defined only for n >1)
282  if (sigma < 0.) return 0.;
283  if ( n <= 1) return std::numeric_limits<double>::quiet_NaN(); // pdf is not normalized for n <=1
284  double abs_alpha = std::abs(alpha);
285  double C = n/abs_alpha * 1./(n-1.) * std::exp(-alpha*alpha/2.);
286  double D = std::sqrt(M_PI/2.)*(1.+ROOT::Math::erf(abs_alpha/std::sqrt(2.)));
287  double N = 1./(sigma*(C+D));
288  return N * crystalball_function(x,alpha,n,sigma,mean);
289  }
290 
291  /**
292 
293  Probability density function of the exponential distribution.
294 
295  \f[ p(x) = \lambda e^{-\lambda x} \f]
296 
297  for x>0. For detailed description see
298  <A HREF="http://mathworld.wolfram.com/ExponentialDistribution.html">
299  Mathworld</A>.
300 
301 
302  @ingroup PdfFunc
303 
304  */
305 
306  inline double exponential_pdf(double x, double lambda, double x0 = 0) {
307  // Inlined to enable clad-auto-derivation for this function.
308 
309  if ((x-x0) < 0)
310  return 0.0;
311  return lambda * std::exp (-lambda * (x-x0));
312  }
313 
314 
315 
316 
317  /**
318 
319  Probability density function of the F-distribution.
320 
321  \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]
322 
323  for x>=0. For detailed description see
324  <A HREF="http://mathworld.wolfram.com/F-Distribution.html">
325  Mathworld</A>.
326 
327  @ingroup PdfFunc
328 
329  */
330 
331 
332  inline double fdistribution_pdf(double x, double n, double m, double x0 = 0) {
333  // Inlined to enable clad-auto-derivation for this function.
334 
335  // function is defined only for both n and m > 0
336  if (n < 0 || m < 0)
337  return std::numeric_limits<double>::quiet_NaN();
338  if ((x-x0) < 0)
339  return 0.0;
340 
341  return std::exp((n/2) * std::log(n) + (m/2) * std::log(m) + ROOT::Math::lgamma((n+m)/2) - ROOT::Math::lgamma(n/2) - ROOT::Math::lgamma(m/2)
342  + (n/2 -1) * std::log(x-x0) - ((n+m)/2) * std::log(m + n*(x-x0)) );
343 
344  }
345 
346 
347 
348 
349  /**
350 
351  Probability density function of the gamma distribution.
352 
353  \f[ p(x) = {1 \over \Gamma(\alpha) \theta^{\alpha}} x^{\alpha-1} e^{-x/\theta} \f]
354 
355  for x>0. For detailed description see
356  <A HREF="http://mathworld.wolfram.com/GammaDistribution.html">
357  Mathworld</A>.
358 
359  @ingroup PdfFunc
360 
361  */
362 
363  inline double gamma_pdf(double x, double alpha, double theta, double x0 = 0) {
364  // Inlined to enable clad-auto-derivation for this function.
365 
366  if ((x-x0) < 0) {
367  return 0.0;
368  } else if ((x-x0) == 0) {
369 
370  if (alpha == 1) {
371  return 1.0/theta;
372  } else {
373  return 0.0;
374  }
375 
376  } else if (alpha == 1) {
377  return std::exp(-(x-x0)/theta)/theta;
378  } else {
379  return std::exp((alpha - 1) * std::log((x-x0)/theta) - (x-x0)/theta - ROOT::Math::lgamma(alpha))/theta;
380  }
381 
382  }
383 
384 
385 
386 
387  /**
388 
389  Probability density function of the normal (Gaussian) distribution.
390 
391  \f[ p(x) = {1 \over \sqrt{2 \pi \sigma^2}} e^{-x^2 / 2\sigma^2} \f]
392 
393  For detailed description see
394  <A HREF="http://mathworld.wolfram.com/NormalDistribution.html">
395  Mathworld</A>. It can also be evaluated using #normal_pdf which will
396  call the same implementation.
397 
398  @ingroup PdfFunc
399 
400  */
401 
402  inline double gaussian_pdf(double x, double sigma = 1, double x0 = 0) {
403 
404  double tmp = (x-x0)/sigma;
405  return (1.0/(std::sqrt(2 * M_PI) * std::fabs(sigma))) * std::exp(-tmp*tmp/2);
406  }
407 
408  /**
409 
410  Probability density function of the bi-dimensional (Gaussian) distribution.
411 
412  \f[ p(x) = {1 \over 2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp (-(x^2/\sigma_x^2 + y^2/\sigma_y^2 - 2 \rho x y/(\sigma_x\sigma_y))/2(1-\rho^2)) \f]
413 
414  For detailed description see
415  <A HREF="http://mathworld.wolfram.com/BivariateNormalDistribution.html">
416  Mathworld</A>. It can also be evaluated using #normal_pdf which will
417  call the same implementation.
418 
419  @param rho correlation , must be between -1,1
420 
421  @ingroup PdfFunc
422 
423  */
424 
425  inline double bigaussian_pdf(double x, double y, double sigmax = 1, double sigmay = 1, double rho = 0, double x0 = 0, double y0 = 0) {
426  double u = (x-x0)/sigmax;
427  double v = (y-y0)/sigmay;
428  double c = 1. - rho*rho;
429  double z = u*u - 2.*rho*u*v + v*v;
430  return 1./(2 * M_PI * sigmax * sigmay * std::sqrt(c) ) * std::exp(- z / (2. * c) );
431  }
432 
433  /**
434 
435  Probability density function of the Landau distribution:
436  \f[ p(x) = \frac{1}{\xi} \phi (\lambda) \f]
437  with
438  \f[ \phi(\lambda) = \frac{1}{2 \pi i}\int_{c-i\infty}^{c+i\infty} e^{\lambda s + s \log{s}} ds\f]
439  where \f$\lambda = (x-x_0)/\xi\f$. For a detailed description see
440  K.S. K&ouml;lbig and B. Schorr, A program package for the Landau distribution,
441  <A HREF="http://dx.doi.org/10.1016/0010-4655(84)90085-7">Computer Phys. Comm. 31 (1984) 97-111</A>
442  <A HREF="http://dx.doi.org/10.1016/j.cpc.2008.03.002">[Erratum-ibid. 178 (2008) 972]</A>.
443  The same algorithms as in
444  <A HREF="https://cern-tex.web.cern.ch/cern-tex/shortwrupsdir/g110/top.html">
445  CERNLIB</A> (DENLAN) is used
446 
447  @param x The argument \f$x\f$
448  @param xi The width parameter \f$\xi\f$
449  @param x0 The location parameter \f$x_0\f$
450 
451  @ingroup PdfFunc
452 
453  */
454 
455  double landau_pdf(double x, double xi = 1, double x0 = 0);
456 
457 
458 
459  /**
460 
461  Probability density function of the lognormal distribution.
462 
463  \f[ p(x) = {1 \over x \sqrt{2 \pi s^2} } e^{-(\ln{x} - m)^2/2 s^2} \f]
464 
465  for x>0. For detailed description see
466  <A HREF="http://mathworld.wolfram.com/LogNormalDistribution.html">
467  Mathworld</A>.
468  @param s scale parameter (not the sigma of the distribution which is not even defined)
469  @param x0 location parameter, corresponds approximately to the most probable value. For x0 = 0, sigma = 1, the x_mpv = -0.22278
470 
471  @ingroup PdfFunc
472 
473  */
474 
475  inline double lognormal_pdf(double x, double m, double s, double x0 = 0) {
476  // Inlined to enable clad-auto-derivation for this function.
477  if ((x-x0) <= 0)
478  return 0.0;
479  double tmp = (std::log((x-x0)) - m)/s;
480  return 1.0 / ((x-x0) * std::fabs(s) * std::sqrt(2 * M_PI)) * std::exp(-(tmp * tmp) /2);
481  }
482 
483 
484 
485 
486  /**
487 
488  Probability density function of the normal (Gaussian) distribution.
489 
490  \f[ p(x) = {1 \over \sqrt{2 \pi \sigma^2}} e^{-x^2 / 2\sigma^2} \f]
491 
492  For detailed description see
493  <A HREF="http://mathworld.wolfram.com/NormalDistribution.html">
494  Mathworld</A>. It can also be evaluated using #gaussian_pdf which will call the same
495  implementation.
496 
497  @ingroup PdfFunc
498 
499  */
500 
501  inline double normal_pdf(double x, double sigma =1, double x0 = 0) {
502  // Inlined to enable clad-auto-derivation for this function.
503 
504  double tmp = (x-x0)/sigma;
505  return (1.0/(std::sqrt(2 * M_PI) * std::fabs(sigma))) * std::exp(-tmp*tmp/2);
506 
507  }
508 
509 
510  /**
511 
512  Probability density function of the Poisson distribution.
513 
514  \f[ p(n) = \frac{\mu^n}{n!} e^{- \mu} \f]
515 
516  For detailed description see
517  <A HREF="http://mathworld.wolfram.com/PoissonDistribution.html">
518  Mathworld</A>.
519 
520  @ingroup PdfFunc
521 
522  */
523 
524  inline double poisson_pdf(unsigned int n, double mu) {
525  // Inlined to enable clad-auto-derivation for this function.
526 
527  if (n > 0 && mu >= 0)
528  return std::exp (n*std::log(mu) - ROOT::Math::lgamma(n+1) - mu);
529 
530  // when n = 0 and mu = 0, 1 is returned
531  if (mu >= 0)
532  return std::exp(-mu);
533 
534  // return a nan for mu < 0 since it does not make sense
535  return std::numeric_limits<double>::quiet_NaN();
536  }
537 
538 
539 
540 
541  /**
542 
543  Probability density function of Student's t-distribution.
544 
545  \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]
546 
547  for \f$k \geq 0\f$. For detailed description see
548  <A HREF="http://mathworld.wolfram.com/Studentst-Distribution.html">
549  Mathworld</A>.
550 
551  @ingroup PdfFunc
552 
553  */
554 
555  inline double tdistribution_pdf(double x, double r, double x0 = 0) {
556  // Inlined to enable clad-auto-derivation for this function.
557 
558  return (std::exp (ROOT::Math::lgamma((r + 1.0)/2.0) - ROOT::Math::lgamma(r/2.0)) / std::sqrt (M_PI * r))
559  * std::pow ((1.0 + (x-x0)*(x-x0)/r), -(r + 1.0)/2.0);
560 
561  }
562 
563 
564 
565 
566  /**
567 
568  Probability density function of the uniform (flat) distribution.
569 
570  \f[ p(x) = {1 \over (b-a)} \f]
571 
572  if \f$a \leq x<b\f$ and 0 otherwise. For detailed description see
573  <A HREF="http://mathworld.wolfram.com/UniformDistribution.html">
574  Mathworld</A>.
575 
576  @ingroup PdfFunc
577 
578  */
579 
580  inline double uniform_pdf(double x, double a, double b, double x0 = 0) {
581  // Inlined to enable clad-auto-derivation for this function.
582 
583  //!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! when a=b
584 
585  if ((x-x0) < b && (x-x0) >= a)
586  return 1.0/(b - a);
587  return 0.0;
588 
589  }
590 
591 
592 
593 } // namespace Math
594 } // namespace ROOT
595 
596 
597 
598 #endif // ROOT_Math_PdfFunc
c
#define c(i)
Definition: RSha256.hxx:119
ROOT::Math::Cephes::gamma
double gamma(double x)
Definition: SpecFuncCephes.cxx:339
m
auto * m
Definition: textangle.C:8
n
const Int_t n
Definition: legend1.C:16
ROOT::Math::exponential_pdf
double exponential_pdf(double x, double lambda, double x0=0)
Probability density function of the exponential distribution.
Definition: PdfFuncMathCore.h:306
ROOT::Math::erf
double erf(double x)
Error function encountered in integrating the normal distribution.
Definition: SpecFuncMathCore.cxx:59
ROOT::Math::beta_pdf
double beta_pdf(double x, double a, double b)
Probability density function of the beta distribution.
Definition: PdfFuncMathCore.h:82
r
ROOT::R::TRInterface & r
Definition: Object.C:4
exp
double exp(double)
log
double log(double)
TGeant4Unit::s
static constexpr double s
Definition: TGeant4SystemOfUnits.h:168
N
#define N
x
Double_t x[n]
Definition: legend1.C:17
SpecFuncMathCore.h
v
@ v
Definition: rootcling_impl.cxx:3635
b
#define b(i)
Definition: RSha256.hxx:118
ROOT::Math::bigaussian_pdf
double 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.
Definition: PdfFuncMathCore.h:425
ROOT::Math::binomial_pdf
double binomial_pdf(unsigned int k, double p, unsigned int n)
Probability density function of the binomial distribution.
Definition: PdfFuncMathCore.h:118
ROOT::Math::crystalball_pdf
double 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
Definition: PdfFuncMathCore.h:278
ROOT::Math::Cephes::C
static double C[]
Definition: SpecFuncCephes.cxx:187
ROOT::Math::fabs
VecExpr< UnaryOp< Fabs< T >, VecExpr< A, T, D >, T >, T, D > fabs(const VecExpr< A, T, D > &rhs)
Definition: UnaryOperators.h:131
Math.h
ROOT::Math::normal_pdf
double normal_pdf(double x, double sigma=1, double x0=0)
Probability density function of the normal (Gaussian) distribution.
Definition: PdfFuncMathCore.h:501
ROOT::Math::gaussian_pdf
double gaussian_pdf(double x, double sigma=1, double x0=0)
Probability density function of the normal (Gaussian) distribution.
Definition: PdfFuncMathCore.h:402
ROOT::Math::gamma_pdf
double gamma_pdf(double x, double alpha, double theta, double x0=0)
Probability density function of the gamma distribution.
Definition: PdfFuncMathCore.h:363
ROOT::Math::breitwigner_pdf
double breitwigner_pdf(double x, double gamma, double x0=0)
Probability density function of Breit-Wigner distribution, which is similar, just a different definit...
Definition: PdfFuncMathCore.h:175
ROOT::Math::landau_pdf
double landau_pdf(double x, double xi=1, double x0=0)
Probability density function of the Landau distribution:
Definition: PdfFuncMathCore.cxx:31
a
auto * a
Definition: textangle.C:12
ROOT::Math::cauchy_pdf
double cauchy_pdf(double x, double b=1, double x0=0)
Probability density function of the Cauchy distribution which is also called Lorentzian distribution.
Definition: PdfFuncMathCore.h:201
ROOT::Math::crystalball_function
double crystalball_function(double x, double alpha, double n, double sigma, double mean=0)
Crystal ball function.
Definition: PdfFuncMathCore.h:254
y
Double_t y[n]
Definition: legend1.C:17
ROOT::Math::poisson_pdf
double poisson_pdf(unsigned int n, double mu)
Probability density function of the Poisson distribution.
Definition: PdfFuncMathCore.h:524
sqrt
double sqrt(double)
ROOT::Math::negative_binomial_pdf
double negative_binomial_pdf(unsigned int k, double p, double n)
Probability density function of the negative binomial distribution.
Definition: PdfFuncMathCore.h:146
ROOT::Math::fdistribution_pdf
double fdistribution_pdf(double x, double n, double m, double x0=0)
Probability density function of the F-distribution.
Definition: PdfFuncMathCore.h:332
sigma
const Double_t sigma
Definition: h1analysisProxy.h:11
M_PI
#define M_PI
Definition: Rotated.cxx:105
ROOT::Math::log1p
double log1p(double x)
declarations for functions which are not implemented by some compilers
Definition: Math.h:98
ROOT::Math::lognormal_pdf
double lognormal_pdf(double x, double m, double s, double x0=0)
Probability density function of the lognormal distribution.
Definition: PdfFuncMathCore.h:475
ROOT::Math::lgamma
double lgamma(double x)
Calculates the logarithm of the gamma function.
Definition: SpecFuncMathCore.cxx:74
ROOT::Math::Cephes::B
static double B[]
Definition: SpecFuncCephes.cxx:178
ROOT::Math::chisquared_pdf
double chisquared_pdf(double x, double r, double x0=0)
Probability density function of the distribution with degrees of freedom.
Definition: PdfFuncMathCore.h:225
pow
double pow(double, double)
ROOT
VSD Structures.
Definition: StringConv.hxx:21
Math
ROOT::Math::uniform_pdf
double uniform_pdf(double x, double a, double b, double x0=0)
Probability density function of the uniform (flat) distribution.
Definition: PdfFuncMathCore.h:580
ROOT::Math::tdistribution_pdf
double tdistribution_pdf(double x, double r, double x0=0)
Probability density function of Student's t-distribution.
Definition: PdfFuncMathCore.h:555