Logo ROOT   6.19/01
Reference Guide
ProbFuncMathCore.h
Go to the documentation of this file.
1 // @(#)root/mathcore:$Id$
2 // Authors: L. Moneta, A. Zsenei 06/2005
3 
4 /**********************************************************************
5  * *
6  * Copyright (c) 2005 , LCG ROOT MathLib Team *
7  * *
8  * *
9  **********************************************************************/
10 
11 #ifndef ROOT_Math_ProbFuncMathCore
12 #define ROOT_Math_ProbFuncMathCore
13 
14 
15 namespace ROOT {
16 namespace Math {
17 
18 
19  /** @defgroup ProbFunc Cumulative Distribution Functions (CDF)
20 
21  @ingroup StatFunc
22 
23  * Cumulative distribution functions of various distributions.
24  * The functions with the extension <em>_cdf</em> calculate the
25  * lower tail integral of the probability density function
26  *
27  * \f[ D(x) = \int_{-\infty}^{x} p(x') dx' \f]
28  *
29  * while those with the <em>_cdf_c</em> extension calculate the complement of
30  * cumulative distribution function, called in statistics the survival
31  * function.
32  * It corresponds to the upper tail integral of the
33  * probability density function
34  *
35  * \f[ D(x) = \int_{x}^{+\infty} p(x') dx' \f]
36  *
37  *
38  * <strong>NOTE:</strong> In the old releases (< 5.14) the <em>_cdf</em> functions were called
39  * <em>_quant</em> and the <em>_cdf_c</em> functions were called
40  * <em>_prob</em>.
41  * These names are currently kept for backward compatibility, but
42  * their usage is deprecated.
43  *
44  * These functions are defined in the header file <em>Math/ProbFunc.h<em> or in the global one
45  * including all statistical functions <em>Math/DistFunc.h<em>
46  *
47  */
48 
49 
50 
51  /**
52 
53  Complement of the cumulative distribution function of the beta distribution.
54  Upper tail of the integral of the #beta_pdf
55 
56  @ingroup ProbFunc
57 
58  */
59 
60  double beta_cdf_c(double x, double a, double b);
61 
62 
63 
64  /**
65 
66  Cumulative distribution function of the beta distribution
67  Upper tail of the integral of the #beta_pdf
68 
69  @ingroup ProbFunc
70 
71  */
72 
73  double beta_cdf(double x, double a, double b);
74 
75 
76 
77 
78  /**
79 
80  Complement of the cumulative distribution function (upper tail) of the Breit_Wigner
81  distribution and it is similar (just a different parameter definition) to the
82  Cauchy distribution (see #cauchy_cdf_c )
83 
84  \f[ D(x) = \int_{x}^{+\infty} \frac{1}{\pi} \frac{\frac{1}{2} \Gamma}{x'^2 + (\frac{1}{2} \Gamma)^2} dx' \f]
85 
86 
87  @ingroup ProbFunc
88 
89  */
90  double breitwigner_cdf_c(double x, double gamma, double x0 = 0);
91 
92 
93  /**
94 
95  Cumulative distribution function (lower tail) of the Breit_Wigner
96  distribution and it is similar (just a different parameter definition) to the
97  Cauchy distribution (see #cauchy_cdf )
98 
99  \f[ D(x) = \int_{-\infty}^{x} \frac{1}{\pi} \frac{b}{x'^2 + (\frac{1}{2} \Gamma)^2} dx' \f]
100 
101 
102  @ingroup ProbFunc
103 
104  */
105  double breitwigner_cdf(double x, double gamma, double x0 = 0);
106 
107 
108 
109  /**
110 
111  Complement of the cumulative distribution function (upper tail) of the
112  Cauchy distribution which is also Lorentzian distribution.
113  It is similar (just a different parameter definition) to the
114  Breit_Wigner distribution (see #breitwigner_cdf_c )
115 
116  \f[ D(x) = \int_{x}^{+\infty} \frac{1}{\pi} \frac{ b }{ (x'-m)^2 + b^2} dx' \f]
117 
118  For detailed description see
119  <A HREF="http://mathworld.wolfram.com/CauchyDistribution.html">
120  Mathworld</A>.
121 
122  @ingroup ProbFunc
123 
124  */
125  double cauchy_cdf_c(double x, double b, double x0 = 0);
126 
127 
128 
129 
130  /**
131 
132  Cumulative distribution function (lower tail) of the
133  Cauchy distribution which is also Lorentzian distribution.
134  It is similar (just a different parameter definition) to the
135  Breit_Wigner distribution (see #breitwigner_cdf )
136 
137  \f[ D(x) = \int_{-\infty}^{x} \frac{1}{\pi} \frac{ b }{ (x'-m)^2 + b^2} dx' \f]
138 
139  For detailed description see
140  <A HREF="http://mathworld.wolfram.com/CauchyDistribution.html">
141  Mathworld</A>.
142 
143 
144  @ingroup ProbFunc
145 
146  */
147  double cauchy_cdf(double x, double b, double x0 = 0);
148 
149 
150 
151 
152  /**
153 
154  Complement of the cumulative distribution function of the \f$\chi^2\f$ distribution
155  with \f$r\f$ degrees of freedom (upper tail).
156 
157  \f[ D_{r}(x) = \int_{x}^{+\infty} \frac{1}{\Gamma(r/2) 2^{r/2}} x'^{r/2-1} e^{-x'/2} dx' \f]
158 
159  For detailed description see
160  <A HREF="http://mathworld.wolfram.com/Chi-SquaredDistribution.html">
161  Mathworld</A>. It is implemented using the incomplete gamma function, ROOT::Math::inc_gamma_c,
162  from <A HREF="http://www.netlib.org/cephes">Cephes</A>
163 
164  @ingroup ProbFunc
165 
166  */
167 
168  double chisquared_cdf_c(double x, double r, double x0 = 0);
169 
170 
171 
172  /**
173 
174  Cumulative distribution function of the \f$\chi^2\f$ distribution
175  with \f$r\f$ degrees of freedom (lower tail).
176 
177  \f[ D_{r}(x) = \int_{-\infty}^{x} \frac{1}{\Gamma(r/2) 2^{r/2}} x'^{r/2-1} e^{-x'/2} dx' \f]
178 
179  For detailed description see
180  <A HREF="http://mathworld.wolfram.com/Chi-SquaredDistribution.html">
181  Mathworld</A>. It is implemented using the incomplete gamma function, ROOT::Math::inc_gamma_c,
182  from <A HREF="http://www.netlib.org/cephes">Cephes</A>
183 
184  @ingroup ProbFunc
185 
186  */
187 
188  double chisquared_cdf(double x, double r, double x0 = 0);
189 
190 
191  /**
192 
193  Cumulative distribution for the Crystal Ball distribution function
194 
195  See the definition of the Crystal Ball function at
196  <A HREF="http://en.wikipedia.org/wiki/Crystal_Ball_function">
197  Wikipedia</A>.
198 
199  The distribution is defined only for n > 1 when the integral converges
200 
201  @ingroup ProbFunc
202 
203  */
204  double crystalball_cdf(double x, double alpha, double n, double sigma, double x0 = 0);
205 
206  /**
207 
208  Complement of the Cumulative distribution for the Crystal Ball distribution
209 
210  See the definition of the Crystal Ball function at
211  <A HREF="http://en.wikipedia.org/wiki/Crystal_Ball_function">
212  Wikipedia</A>.
213 
214  The distribution is defined only for n > 1 when the integral converges
215 
216  @ingroup ProbFunc
217 
218  */
219  double crystalball_cdf_c(double x, double alpha, double n, double sigma, double x0 = 0);
220 
221  /**
222  Integral of the not-normalized Crystal Ball function
223 
224  See the definition of the Crystal Ball function at
225  <A HREF="http://en.wikipedia.org/wiki/Crystal_Ball_function">
226  Wikipedia</A>.
227 
228  see ROOT::Math::crystalball_function for the function evaluation.
229 
230  @ingroup ProbFunc
231 
232  */
233  double crystalball_integral(double x, double alpha, double n, double sigma, double x0 = 0);
234 
235  /**
236 
237  Complement of the cumulative distribution function of the exponential distribution
238  (upper tail).
239 
240  \f[ D(x) = \int_{x}^{+\infty} \lambda e^{-\lambda x'} dx' \f]
241 
242  For detailed description see
243  <A HREF="http://mathworld.wolfram.com/ExponentialDistribution.html">
244  Mathworld</A>.
245 
246  @ingroup ProbFunc
247 
248  */
249 
250  double exponential_cdf_c(double x, double lambda, double x0 = 0);
251 
252 
253 
254  /**
255 
256  Cumulative distribution function of the exponential distribution
257  (lower tail).
258 
259  \f[ D(x) = \int_{-\infty}^{x} \lambda e^{-\lambda x'} dx' \f]
260 
261  For detailed description see
262  <A HREF="http://mathworld.wolfram.com/ExponentialDistribution.html">
263  Mathworld</A>.
264 
265  @ingroup ProbFunc
266 
267  */
268 
269 
270  double exponential_cdf(double x, double lambda, double x0 = 0);
271 
272 
273 
274  /**
275 
276  Complement of the cumulative distribution function of the F-distribution
277  (upper tail).
278 
279  \f[ 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' \f]
280 
281  For detailed description see
282  <A HREF="http://mathworld.wolfram.com/F-Distribution.html">
283  Mathworld</A>. It is implemented using the incomplete beta function, ROOT::Math::inc_beta,
284  from <A HREF="http://www.netlib.org/cephes">Cephes</A>
285 
286  @ingroup ProbFunc
287 
288  */
289 
290  double fdistribution_cdf_c(double x, double n, double m, double x0 = 0);
291 
292 
293 
294 
295  /**
296 
297  Cumulative distribution function of the F-distribution
298  (lower tail).
299 
300  \f[ 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' \f]
301 
302  For detailed description see
303  <A HREF="http://mathworld.wolfram.com/F-Distribution.html">
304  Mathworld</A>. It is implemented using the incomplete beta function, ROOT::Math::inc_beta,
305  from <A HREF="http://www.netlib.org/cephes">Cephes</A>
306 
307  @ingroup ProbFunc
308 
309  */
310 
311  double fdistribution_cdf(double x, double n, double m, double x0 = 0);
312 
313 
314 
315  /**
316 
317  Complement of the cumulative distribution function of the gamma distribution
318  (upper tail).
319 
320  \f[ D(x) = \int_{x}^{+\infty} {1 \over \Gamma(\alpha) \theta^{\alpha}} x'^{\alpha-1} e^{-x'/\theta} dx' \f]
321 
322  For detailed description see
323  <A HREF="http://mathworld.wolfram.com/GammaDistribution.html">
324  Mathworld</A>. It is implemented using the incomplete gamma function, ROOT::Math::inc_gamma,
325  from <A HREF="http://www.netlib.org/cephes">Cephes</A>
326 
327  @ingroup ProbFunc
328 
329  */
330 
331  double gamma_cdf_c(double x, double alpha, double theta, double x0 = 0);
332 
333 
334 
335 
336  /**
337 
338  Cumulative distribution function of the gamma distribution
339  (lower tail).
340 
341  \f[ D(x) = \int_{-\infty}^{x} {1 \over \Gamma(\alpha) \theta^{\alpha}} x'^{\alpha-1} e^{-x'/\theta} dx' \f]
342 
343  For detailed description see
344  <A HREF="http://mathworld.wolfram.com/GammaDistribution.html">
345  Mathworld</A>. It is implemented using the incomplete gamma function, ROOT::Math::inc_gamma,
346  from <A HREF="http://www.netlib.org/cephes">Cephes</A>
347 
348  @ingroup ProbFunc
349 
350  */
351 
352  double gamma_cdf(double x, double alpha, double theta, double x0 = 0);
353 
354 
355 
356  /**
357 
358  Cumulative distribution function of the Landau
359  distribution (lower tail).
360 
361  \f[ D(x) = \int_{-\infty}^{x} p(x) dx \f]
362 
363  where \f$p(x)\f$ is the Landau probability density function :
364  \f[ p(x) = \frac{1}{\xi} \phi (\lambda) \f]
365  with
366  \f[ \phi(\lambda) = \frac{1}{2 \pi i}\int_{c-i\infty}^{c+i\infty} e^{\lambda s + s \log{s}} ds\f]
367  with \f$\lambda = (x-x_0)/\xi\f$. For a detailed description see
368  K.S. K&ouml;lbig and B. Schorr, A program package for the Landau distribution,
369  <A HREF="http://dx.doi.org/10.1016/0010-4655(84)90085-7">Computer Phys. Comm. 31 (1984) 97-111</A>
370  <A HREF="http://dx.doi.org/10.1016/j.cpc.2008.03.002">[Erratum-ibid. 178 (2008) 972]</A>.
371  The same algorithms as in
372  <A HREF="https://cern-tex.web.cern.ch/cern-tex/shortwrupsdir/g110/top.html">
373  CERNLIB</A> (DISLAN) is used.
374 
375  @param x The argument \f$x\f$
376  @param xi The width parameter \f$\xi\f$
377  @param x0 The location parameter \f$x_0\f$
378 
379  @ingroup ProbFunc
380 
381  */
382 
383  double landau_cdf(double x, double xi = 1, double x0 = 0);
384 
385  /**
386 
387  Complement of the distribution function of the Landau
388  distribution (upper tail).
389 
390  \f[ D(x) = \int_{x}^{+\infty} p(x) dx \f]
391 
392  where p(x) is the Landau probability density function.
393  It is implemented simply as 1. - #landau_cdf
394 
395  @param x The argument \f$x\f$
396  @param xi The width parameter \f$\xi\f$
397  @param x0 The location parameter \f$x_0\f$
398 
399  @ingroup ProbFunc
400 
401  */
402  inline double landau_cdf_c(double x, double xi = 1, double x0 = 0) {
403  return 1. - landau_cdf(x,xi,x0);
404  }
405 
406  /**
407 
408  Complement of the cumulative distribution function of the lognormal distribution
409  (upper tail).
410 
411  \f[ D(x) = \int_{x}^{+\infty} {1 \over x' \sqrt{2 \pi s^2} } e^{-(\ln{x'} - m)^2/2 s^2} dx' \f]
412 
413  For detailed description see
414  <A HREF="http://mathworld.wolfram.com/LogNormalDistribution.html">
415  Mathworld</A>.
416 
417  @ingroup ProbFunc
418 
419  */
420 
421  double lognormal_cdf_c(double x, double m, double s, double x0 = 0);
422 
423 
424 
425 
426  /**
427 
428  Cumulative distribution function of the lognormal distribution
429  (lower tail).
430 
431  \f[ D(x) = \int_{-\infty}^{x} {1 \over x' \sqrt{2 \pi s^2} } e^{-(\ln{x'} - m)^2/2 s^2} dx' \f]
432 
433  For detailed description see
434  <A HREF="http://mathworld.wolfram.com/LogNormalDistribution.html">
435  Mathworld</A>.
436 
437  @ingroup ProbFunc
438 
439  */
440 
441  double lognormal_cdf(double x, double m, double s, double x0 = 0);
442 
443 
444 
445 
446  /**
447 
448  Complement of the cumulative distribution function of the normal (Gaussian)
449  distribution (upper tail).
450 
451  \f[ D(x) = \int_{x}^{+\infty} {1 \over \sqrt{2 \pi \sigma^2}} e^{-x'^2 / 2\sigma^2} dx' \f]
452 
453  For detailed description see
454  <A HREF="http://mathworld.wolfram.com/NormalDistribution.html">
455  Mathworld</A>.
456 
457  @ingroup ProbFunc
458 
459  */
460 
461  double normal_cdf_c(double x, double sigma = 1, double x0 = 0);
462  /// Alternative name for same function
463  inline double gaussian_cdf_c(double x, double sigma = 1, double x0 = 0) {
464  return normal_cdf_c(x,sigma,x0);
465  }
466 
467 
468 
469  /**
470 
471  Cumulative distribution function of the normal (Gaussian)
472  distribution (lower tail).
473 
474  \f[ D(x) = \int_{-\infty}^{x} {1 \over \sqrt{2 \pi \sigma^2}} e^{-x'^2 / 2\sigma^2} dx' \f]
475 
476  For detailed description see
477  <A HREF="http://mathworld.wolfram.com/NormalDistribution.html">
478  Mathworld</A>.
479  @ingroup ProbFunc
480 
481  */
482 
483  double normal_cdf(double x, double sigma = 1, double x0 = 0);
484  /// Alternative name for same function
485  inline double gaussian_cdf(double x, double sigma = 1, double x0 = 0) {
486  return normal_cdf(x,sigma,x0);
487  }
488 
489 
490 
491  /**
492 
493  Complement of the cumulative distribution function of Student's
494  t-distribution (upper tail).
495 
496  \f[ 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' \f]
497 
498  For detailed description see
499  <A HREF="http://mathworld.wolfram.com/Studentst-Distribution.html">
500  Mathworld</A>. It is implemented using the incomplete beta function, ROOT::Math::inc_beta,
501  from <A HREF="http://www.netlib.org/cephes">Cephes</A>
502 
503  @ingroup ProbFunc
504 
505  */
506 
507  double tdistribution_cdf_c(double x, double r, double x0 = 0);
508 
509 
510 
511 
512  /**
513 
514  Cumulative distribution function of Student's
515  t-distribution (lower tail).
516 
517  \f[ 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' \f]
518 
519  For detailed description see
520  <A HREF="http://mathworld.wolfram.com/Studentst-Distribution.html">
521  Mathworld</A>. It is implemented using the incomplete beta function, ROOT::Math::inc_beta,
522  from <A HREF="http://www.netlib.org/cephes">Cephes</A>
523 
524  @ingroup ProbFunc
525 
526  */
527 
528  double tdistribution_cdf(double x, double r, double x0 = 0);
529 
530 
531  /**
532 
533  Complement of the cumulative distribution function of the uniform (flat)
534  distribution (upper tail).
535 
536  \f[ D(x) = \int_{x}^{+\infty} {1 \over (b-a)} dx' \f]
537 
538  For detailed description see
539  <A HREF="http://mathworld.wolfram.com/UniformDistribution.html">
540  Mathworld</A>.
541 
542  @ingroup ProbFunc
543 
544  */
545 
546  double uniform_cdf_c(double x, double a, double b, double x0 = 0);
547 
548 
549 
550 
551  /**
552 
553  Cumulative distribution function of the uniform (flat)
554  distribution (lower tail).
555 
556  \f[ D(x) = \int_{-\infty}^{x} {1 \over (b-a)} dx' \f]
557 
558  For detailed description see
559  <A HREF="http://mathworld.wolfram.com/UniformDistribution.html">
560  Mathworld</A>.
561 
562  @ingroup ProbFunc
563 
564  */
565 
566  double uniform_cdf(double x, double a, double b, double x0 = 0);
567 
568 
569 
570 
571  /**
572 
573  Complement of the cumulative distribution function of the Poisson distribution.
574  Upper tail of the integral of the #poisson_pdf
575 
576  @ingroup ProbFunc
577 
578  */
579 
580  double poisson_cdf_c(unsigned int n, double mu);
581 
582  /**
583 
584  Cumulative distribution function of the Poisson distribution
585  Lower tail of the integral of the #poisson_pdf
586 
587  @ingroup ProbFunc
588 
589  */
590 
591  double poisson_cdf(unsigned int n, double mu);
592 
593  /**
594 
595  Complement of the cumulative distribution function of the Binomial distribution.
596  Upper tail of the integral of the #binomial_pdf
597 
598  @ingroup ProbFunc
599 
600  */
601 
602  double binomial_cdf_c(unsigned int k, double p, unsigned int n);
603 
604  /**
605 
606  Cumulative distribution function of the Binomial distribution
607  Lower tail of the integral of the #binomial_pdf
608 
609  @ingroup ProbFunc
610 
611  */
612 
613  double binomial_cdf(unsigned int k, double p, unsigned int n);
614 
615 
616  /**
617 
618  Complement of the cumulative distribution function of the Negative Binomial distribution.
619  Upper tail of the integral of the #negative_binomial_pdf
620 
621  @ingroup ProbFunc
622 
623  */
624 
625  double negative_binomial_cdf_c(unsigned int k, double p, double n);
626 
627  /**
628 
629  Cumulative distribution function of the Negative Binomial distribution
630  Lower tail of the integral of the #negative_binomial_pdf
631 
632  @ingroup ProbFunc
633 
634  */
635 
636  double negative_binomial_cdf(unsigned int k, double p, double n);
637 
638 
639 
640 #ifdef HAVE_OLD_STAT_FUNC
641 
642  /** @name Backward compatible MathCore CDF functions */
643 
644 
645  inline double breitwigner_prob(double x, double gamma, double x0 = 0) {
646  return breitwigner_cdf_c(x,gamma,x0);
647  }
648  inline double breitwigner_quant(double x, double gamma, double x0 = 0) {
649  return breitwigner_cdf(x,gamma,x0);
650  }
651 
652  inline double cauchy_prob(double x, double b, double x0 = 0) {
653  return cauchy_cdf_c(x,b,x0);
654  }
655  inline double cauchy_quant(double x, double b, double x0 = 0) {
656  return cauchy_cdf (x,b,x0);
657  }
658  inline double chisquared_prob(double x, double r, double x0 = 0) {
659  return chisquared_cdf_c(x, r, x0);
660  }
661  inline double chisquared_quant(double x, double r, double x0 = 0) {
662  return chisquared_cdf (x, r, x0);
663  }
664  inline double exponential_prob(double x, double lambda, double x0 = 0) {
665  return exponential_cdf_c(x, lambda, x0 );
666  }
667  inline double exponential_quant(double x, double lambda, double x0 = 0) {
668  return exponential_cdf (x, lambda, x0 );
669  }
670 
671  inline double gaussian_prob(double x, double sigma, double x0 = 0) {
672  return gaussian_cdf_c( x, sigma, x0 );
673  }
674  inline double gaussian_quant(double x, double sigma, double x0 = 0) {
675  return gaussian_cdf ( x, sigma, x0 );
676  }
677 
678  inline double lognormal_prob(double x, double m, double s, double x0 = 0) {
679  return lognormal_cdf_c( x, m, s, x0 );
680  }
681  inline double lognormal_quant(double x, double m, double s, double x0 = 0) {
682  return lognormal_cdf ( x, m, s, x0 );
683  }
684 
685  inline double normal_prob(double x, double sigma, double x0 = 0) {
686  return normal_cdf_c( x, sigma, x0 );
687  }
688  inline double normal_quant(double x, double sigma, double x0 = 0) {
689  return normal_cdf ( x, sigma, x0 );
690  }
691 
692  inline double uniform_prob(double x, double a, double b, double x0 = 0) {
693  return uniform_cdf_c( x, a, b, x0 );
694  }
695  inline double uniform_quant(double x, double a, double b, double x0 = 0) {
696  return uniform_cdf ( x, a, b, x0 );
697  }
698  inline double fdistribution_prob(double x, double n, double m, double x0 = 0) {
699  return fdistribution_cdf_c (x, n, m, x0);
700  }
701  inline double fdistribution_quant(double x, double n, double m, double x0 = 0) {
702  return fdistribution_cdf (x, n, m, x0);
703  }
704 
705  inline double gamma_prob(double x, double alpha, double theta, double x0 = 0) {
706  return gamma_cdf_c (x, alpha, theta, x0);
707  }
708  inline double gamma_quant(double x, double alpha, double theta, double x0 = 0) {
709  return gamma_cdf (x, alpha, theta, x0);
710  }
711 
712  inline double tdistribution_prob(double x, double r, double x0 = 0) {
713  return tdistribution_cdf_c (x, r, x0);
714  }
715 
716  inline double tdistribution_quant(double x, double r, double x0 = 0) {
717  return tdistribution_cdf (x, r, x0);
718  }
719 
720 #endif
721 
722  /** @defgroup TruncFunc Statistical functions from truncated distributions
723 
724  @ingroup StatFunc
725 
726  Statistical functions for the truncated distributions. Examples of such functions are the
727  first or the second momentum of the truncated distribution.
728  In the case of the Landau, first and second momentum functions are provided for the Landau
729  distribution truncated only on the right side.
730  These functions are defined in the header file <em>Math/ProbFunc.h<em> or in the global one
731  including all statistical functions <em>Math/StatFunc.h<em>
732 
733  */
734 
735  /**
736 
737  First moment (mean) of the truncated Landau distribution.
738  \f[ \frac{1}{D (x)} \int_{-\infty}^{x} t\, p(t) d t \f]
739  where \f$p(x)\f$ is the Landau distribution
740  and \f$D(x)\f$ its cumulative distribution function.
741 
742  For detailed description see
743  K.S. K&ouml;lbig and B. Schorr, A program package for the Landau distribution,
744  <A HREF="http://dx.doi.org/10.1016/0010-4655(84)90085-7">Computer Phys. Comm. 31 (1984) 97-111</A>
745  <A HREF="http://dx.doi.org/10.1016/j.cpc.2008.03.002">[Erratum-ibid. 178 (2008) 972]</A>.
746  The same algorithms as in
747  <A HREF="https://cern-tex.web.cern.ch/cern-tex/shortwrupsdir/g110/top.html">
748  CERNLIB</A> (XM1LAN) is used
749 
750  @param x The argument \f$x\f$
751  @param xi The width parameter \f$\xi\f$
752  @param x0 The location parameter \f$x_0\f$
753 
754  @ingroup TruncFunc
755 
756  */
757 
758  double landau_xm1(double x, double xi = 1, double x0 = 0);
759 
760 
761 
762  /**
763 
764  Second moment of the truncated Landau distribution.
765  \f[ \frac{1}{D (x)} \int_{-\infty}^{x} t^2\, p(t) d t \f]
766  where \f$p(x)\f$ is the Landau distribution
767  and \f$D(x)\f$ its cumulative distribution function.
768 
769  For detailed description see
770  K.S. K&ouml;lbig and B. Schorr, A program package for the Landau distribution,
771  <A HREF="http://dx.doi.org/10.1016/0010-4655(84)90085-7">Computer Phys. Comm. 31 (1984) 97-111</A>
772  <A HREF="http://dx.doi.org/10.1016/j.cpc.2008.03.002">[Erratum-ibid. 178 (2008) 972]</A>.
773  The same algorithms as in
774  <A HREF="https://cern-tex.web.cern.ch/cern-tex/shortwrupsdir/g110/top.html">
775  CERNLIB</A> (XM1LAN) is used
776 
777  @param x The argument \f$x\f$
778  @param xi The width parameter \f$\xi\f$
779  @param x0 The location parameter \f$x_0\f$
780 
781  @ingroup TruncFunc
782 
783  */
784 
785  double landau_xm2(double x, double xi = 1, double x0 = 0);
786 
787 
788 
789 } // namespace Math
790 } // namespace ROOT
791 
792 
793 #endif // ROOT_Math_ProbFuncMathCore
double tdistribution_cdf(double x, double r, double x0=0)
Cumulative distribution function of Student&#39;s t-distribution (lower tail).
double binomial_cdf(unsigned int k, double p, unsigned int n)
Cumulative distribution function of the Binomial distribution Lower tail of the integral of the binom...
double landau_cdf_c(double x, double xi=1, double x0=0)
Complement of the distribution function of the Landau distribution (upper tail).
auto * m
Definition: textangle.C:8
VSD Structures.
Definition: StringConv.hxx:21
double crystalball_cdf(double x, double alpha, double n, double sigma, double x0=0)
Cumulative distribution for the Crystal Ball distribution function.
double negative_binomial_cdf_c(unsigned int k, double p, double n)
Complement of the cumulative distribution function of the Negative Binomial distribution.
double poisson_cdf(unsigned int n, double mu)
Cumulative distribution function of the Poisson distribution Lower tail of the integral of the poisso...
double poisson_cdf_c(unsigned int n, double mu)
Complement of the cumulative distribution function of the Poisson distribution.
double binomial_cdf_c(unsigned int k, double p, unsigned int n)
Complement of the cumulative distribution function of the Binomial distribution.
double 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 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 landau_xm1(double x, double xi=1, double x0=0)
First moment (mean) of the truncated Landau distribution.
double lognormal_cdf(double x, double m, double s, double x0=0)
Cumulative distribution function of the lognormal distribution (lower tail).
double negative_binomial_cdf(unsigned int k, double p, double n)
Cumulative distribution function of the Negative Binomial distribution Lower tail of the integral of ...
Double_t x[n]
Definition: legend1.C:17
double 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 normal_cdf(double x, double sigma=1, double x0=0)
Cumulative distribution function of the normal (Gaussian) distribution (lower tail).
double cauchy_cdf_c(double x, double b, double x0=0)
Complement of the cumulative distribution function (upper tail) of the Cauchy distribution which is a...
double fdistribution_cdf(double x, double n, double m, double x0=0)
Cumulative distribution function of the F-distribution (lower tail).
const Double_t sigma
static constexpr double s
double uniform_cdf(double x, double a, double b, double x0=0)
Cumulative distribution function of the uniform (flat) distribution (lower tail). ...
double 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 breitwigner_cdf(double x, double gamma, double x0=0)
Cumulative distribution function (lower tail) of the Breit_Wigner distribution and it is similar (jus...
double landau_cdf(double x, double xi=1, double x0=0)
Cumulative distribution function of the Landau distribution (lower tail).
double landau_xm2(double x, double xi=1, double x0=0)
Second moment of the truncated Landau distribution.
double gamma(double x)
ROOT::R::TRInterface & r
Definition: Object.C:4
double exponential_cdf_c(double x, double lambda, double x0=0)
Complement of the cumulative distribution function of the exponential distribution (upper tail)...
auto * a
Definition: textangle.C:12
double cauchy_cdf(double x, double b, double x0=0)
Cumulative distribution function (lower tail) of the Cauchy distribution which is also Lorentzian dis...
double 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 tdistribution_cdf_c(double x, double r, double x0=0)
Complement of the cumulative distribution function of Student&#39;s t-distribution (upper tail)...
double gaussian_cdf_c(double x, double sigma=1, double x0=0)
Alternative name for same function.
double gaussian_cdf(double x, double sigma=1, double x0=0)
Alternative name for same function.
Namespace for new Math classes and functions.
double beta_cdf_c(double x, double a, double b)
Complement of the cumulative distribution function of the beta distribution.
double gamma_cdf(double x, double alpha, double theta, double x0=0)
Cumulative distribution function of the gamma distribution (lower tail).
double chisquared_cdf_c(double x, double r, double x0=0)
Complement of the cumulative distribution function of the distribution with degrees of freedom (upp...
double crystalball_integral(double x, double alpha, double n, double sigma, double x0=0)
Integral of the not-normalized Crystal Ball function.
you should not use this method at all Int_t Int_t Double_t Double_t Double_t Int_t Double_t Double_t Double_t Double_t b
Definition: TRolke.cxx:630
double chisquared_cdf(double x, double r, double x0=0)
Cumulative distribution function of the distribution with degrees of freedom (lower tail)...
double breitwigner_cdf_c(double x, double gamma, double x0=0)
Complement of the cumulative distribution function (upper tail) of the Breit_Wigner distribution and ...
double crystalball_cdf_c(double x, double alpha, double n, double sigma, double x0=0)
Complement of the Cumulative distribution for the Crystal Ball distribution.
const Int_t n
Definition: legend1.C:16
double 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 exponential_cdf(double x, double lambda, double x0=0)
Cumulative distribution function of the exponential distribution (lower tail).