ROOT   6.14/05 Reference Guide
ProbFuncMathCore.h
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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
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
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
Namespace for new ROOT classes and functions.
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
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.
static constexpr double s
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).