1 // boost\math\distributions\binomial.hpp 2 3 // Copyright John Maddock 2006. 4 // Copyright Paul A. Bristow 2007. 5 6 // Use, modification and distribution are subject to the 7 // Boost Software License, Version 1.0. 8 // (See accompanying file LICENSE_1_0.txt 9 // or copy at http://www.boost.org/LICENSE_1_0.txt) 10 11 // http://en.wikipedia.org/wiki/binomial_distribution 12 13 // Binomial distribution is the discrete probability distribution of 14 // the number (k) of successes, in a sequence of 15 // n independent (yes or no, success or failure) Bernoulli trials. 16 17 // It expresses the probability of a number of events occurring in a fixed time 18 // if these events occur with a known average rate (probability of success), 19 // and are independent of the time since the last event. 20 21 // The number of cars that pass through a certain point on a road during a given period of time. 22 // The number of spelling mistakes a secretary makes while typing a single page. 23 // The number of phone calls at a call center per minute. 24 // The number of times a web server is accessed per minute. 25 // The number of light bulbs that burn out in a certain amount of time. 26 // The number of roadkill found per unit length of road 27 28 // http://en.wikipedia.org/wiki/binomial_distribution 29 30 // Given a sample of N measured values k[i], 31 // we wish to estimate the value of the parameter x (mean) 32 // of the binomial population from which the sample was drawn. 33 // To calculate the maximum likelihood value = 1/N sum i = 1 to N of k[i] 34 35 // Also may want a function for EXACTLY k. 36 37 // And probability that there are EXACTLY k occurrences is 38 // exp(-x) * pow(x, k) / factorial(k) 39 // where x is expected occurrences (mean) during the given interval. 40 // For example, if events occur, on average, every 4 min, 41 // and we are interested in number of events occurring in 10 min, 42 // then x = 10/4 = 2.5 43 44 // http://www.itl.nist.gov/div898/handbook/eda/section3/eda366i.htm 45 46 // The binomial distribution is used when there are 47 // exactly two mutually exclusive outcomes of a trial. 48 // These outcomes are appropriately labeled "success" and "failure". 49 // The binomial distribution is used to obtain 50 // the probability of observing x successes in N trials, 51 // with the probability of success on a single trial denoted by p. 52 // The binomial distribution assumes that p is fixed for all trials. 53 54 // P(x, p, n) = n!/(x! * (n-x)!) * p^x * (1-p)^(n-x) 55 56 // http://mathworld.wolfram.com/BinomialCoefficient.html 57 58 // The binomial coefficient (n; k) is the number of ways of picking 59 // k unordered outcomes from n possibilities, 60 // also known as a combination or combinatorial number. 61 // The symbols _nC_k and (n; k) are used to denote a binomial coefficient, 62 // and are sometimes read as "n choose k." 63 // (n; k) therefore gives the number of k-subsets possible out of a set of n distinct items. 64 65 // For example: 66 // The 2-subsets of {1,2,3,4} are the six pairs {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, and {3,4}, so (4; 2)==6. 67 68 // http://functions.wolfram.com/GammaBetaErf/Binomial/ for evaluation. 69 70 // But note that the binomial distribution 71 // (like others including the poisson, negative binomial & Bernoulli) 72 // is strictly defined as a discrete function: only integral values of k are envisaged. 73 // However because of the method of calculation using a continuous gamma function, 74 // it is convenient to treat it as if a continous function, 75 // and permit non-integral values of k. 76 // To enforce the strict mathematical model, users should use floor or ceil functions 77 // on k outside this function to ensure that k is integral. 78 79 #ifndef BOOST_MATH_SPECIAL_BINOMIAL_HPP 80 #define BOOST_MATH_SPECIAL_BINOMIAL_HPP 81 82 #include <boost/math/distributions/fwd.hpp> 83 #include <boost/math/special_functions/beta.hpp> // for incomplete beta. 84 #include <boost/math/distributions/complement.hpp> // complements 85 #include <boost/math/distributions/detail/common_error_handling.hpp> // error checks 86 #include <boost/math/distributions/detail/inv_discrete_quantile.hpp> // error checks 87 #include <boost/math/special_functions/fpclassify.hpp> // isnan. 88 #include <boost/math/tools/roots.hpp> // for root finding. 89 90 #include <utility> 91 92 namespace boost 93 { 94 namespace math 95 { 96 97 template <class RealType, class Policy> 98 class binomial_distribution; 99 100 namespace binomial_detail{ 101 // common error checking routines for binomial distribution functions: 102 template <class RealType, class Policy> check_N(const char * function,const RealType & N,RealType * result,const Policy & pol)103 inline bool check_N(const char* function, const RealType& N, RealType* result, const Policy& pol) 104 { 105 if((N < 0) || !(boost::math::isfinite)(N)) 106 { 107 *result = policies::raise_domain_error<RealType>( 108 function, 109 "Number of Trials argument is %1%, but must be >= 0 !", N, pol); 110 return false; 111 } 112 return true; 113 } 114 template <class RealType, class Policy> check_success_fraction(const char * function,const RealType & p,RealType * result,const Policy & pol)115 inline bool check_success_fraction(const char* function, const RealType& p, RealType* result, const Policy& pol) 116 { 117 if((p < 0) || (p > 1) || !(boost::math::isfinite)(p)) 118 { 119 *result = policies::raise_domain_error<RealType>( 120 function, 121 "Success fraction argument is %1%, but must be >= 0 and <= 1 !", p, pol); 122 return false; 123 } 124 return true; 125 } 126 template <class RealType, class Policy> check_dist(const char * function,const RealType & N,const RealType & p,RealType * result,const Policy & pol)127 inline bool check_dist(const char* function, const RealType& N, const RealType& p, RealType* result, const Policy& pol) 128 { 129 return check_success_fraction( 130 function, p, result, pol) 131 && check_N( 132 function, N, result, pol); 133 } 134 template <class RealType, class Policy> check_dist_and_k(const char * function,const RealType & N,const RealType & p,RealType k,RealType * result,const Policy & pol)135 inline bool check_dist_and_k(const char* function, const RealType& N, const RealType& p, RealType k, RealType* result, const Policy& pol) 136 { 137 if(check_dist(function, N, p, result, pol) == false) 138 return false; 139 if((k < 0) || !(boost::math::isfinite)(k)) 140 { 141 *result = policies::raise_domain_error<RealType>( 142 function, 143 "Number of Successes argument is %1%, but must be >= 0 !", k, pol); 144 return false; 145 } 146 if(k > N) 147 { 148 *result = policies::raise_domain_error<RealType>( 149 function, 150 "Number of Successes argument is %1%, but must be <= Number of Trials !", k, pol); 151 return false; 152 } 153 return true; 154 } 155 template <class RealType, class Policy> check_dist_and_prob(const char * function,const RealType & N,RealType p,RealType prob,RealType * result,const Policy & pol)156 inline bool check_dist_and_prob(const char* function, const RealType& N, RealType p, RealType prob, RealType* result, const Policy& pol) 157 { 158 if(check_dist(function, N, p, result, pol) && detail::check_probability(function, prob, result, pol) == false) 159 return false; 160 return true; 161 } 162 163 template <class T, class Policy> 164 T inverse_binomial_cornish_fisher(T n, T sf, T p, T q, const Policy& pol) 165 { 166 BOOST_MATH_STD_USING 167 // mean: 168 T m = n * sf; 169 // standard deviation: 170 T sigma = sqrt(n * sf * (1 - sf)); 171 // skewness 172 T sk = (1 - 2 * sf) / sigma; 173 // kurtosis: 174 // T k = (1 - 6 * sf * (1 - sf) ) / (n * sf * (1 - sf)); 175 // Get the inverse of a std normal distribution: 176 T x = boost::math::erfc_inv(p > q ? 2 * q : 2 * p, pol) * constants::root_two<T>(); 177 // Set the sign: 178 if(p < 0.5) 179 x = -x; 180 T x2 = x * x; 181 // w is correction term due to skewness 182 T w = x + sk * (x2 - 1) / 6; 183 /* 184 // Add on correction due to kurtosis. 185 // Disabled for now, seems to make things worse? 186 // 187 if(n >= 10) 188 w += k * x * (x2 - 3) / 24 + sk * sk * x * (2 * x2 - 5) / -36; 189 */ 190 w = m + sigma * w; 191 if(w < tools::min_value<T>()) 192 return sqrt(tools::min_value<T>()); 193 if(w > n) 194 return n; 195 return w; 196 } 197 198 template <class RealType, class Policy> 199 RealType quantile_imp(const binomial_distribution<RealType, Policy>& dist, const RealType& p, const RealType& q) 200 { // Quantile or Percent Point Binomial function. 201 // Return the number of expected successes k, 202 // for a given probability p. 203 // 204 // Error checks: 205 BOOST_MATH_STD_USING // ADL of std names 206 RealType result = 0; 207 RealType trials = dist.trials(); 208 RealType success_fraction = dist.success_fraction(); 209 if(false == binomial_detail::check_dist_and_prob( 210 "boost::math::quantile(binomial_distribution<%1%> const&, %1%)", 211 trials, 212 success_fraction, 213 p, 214 &result, Policy())) 215 { 216 return result; 217 } 218 219 // Special cases: 220 // 221 if(p == 0) 222 { // There may actually be no answer to this question, 223 // since the probability of zero successes may be non-zero, 224 // but zero is the best we can do: 225 return 0; 226 } 227 if(p == 1) 228 { // Probability of n or fewer successes is always one, 229 // so n is the most sensible answer here: 230 return trials; 231 } 232 if (p <= pow(1 - success_fraction, trials)) 233 { // p <= pdf(dist, 0) == cdf(dist, 0) 234 return 0; // So the only reasonable result is zero. 235 } // And root finder would fail otherwise. 236 237 // Solve for quantile numerically: 238 // 239 RealType guess = binomial_detail::inverse_binomial_cornish_fisher(trials, success_fraction, p, q, Policy()); 240 RealType factor = 8; 241 if(trials > 100) 242 factor = 1.01f; // guess is pretty accurate 243 else if((trials > 10) && (trials - 1 > guess) && (guess > 3)) 244 factor = 1.15f; // less accurate but OK. 245 else if(trials < 10) 246 { 247 // pretty inaccurate guess in this area: 248 if(guess > trials / 64) 249 { 250 guess = trials / 4; 251 factor = 2; 252 } 253 else 254 guess = trials / 1024; 255 } 256 else 257 factor = 2; // trials largish, but in far tails. 258 259 typedef typename Policy::discrete_quantile_type discrete_quantile_type; 260 boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>(); 261 return detail::inverse_discrete_quantile( 262 dist, 263 p, 264 q, 265 guess, 266 factor, 267 RealType(1), 268 discrete_quantile_type(), 269 max_iter); 270 } // quantile 271 272 } 273 274 template <class RealType = double, class Policy = policies::policy<> > 275 class binomial_distribution 276 { 277 public: 278 typedef RealType value_type; 279 typedef Policy policy_type; 280 binomial_distribution(RealType n=1,RealType p=0.5)281 binomial_distribution(RealType n = 1, RealType p = 0.5) : m_n(n), m_p(p) 282 { // Default n = 1 is the Bernoulli distribution 283 // with equal probability of 'heads' or 'tails. 284 RealType r; 285 binomial_detail::check_dist( 286 "boost::math::binomial_distribution<%1%>::binomial_distribution", 287 m_n, 288 m_p, 289 &r, Policy()); 290 } // binomial_distribution constructor. 291 success_fraction() const292 RealType success_fraction() const 293 { // Probability. 294 return m_p; 295 } trials() const296 RealType trials() const 297 { // Total number of trials. 298 return m_n; 299 } 300 301 enum interval_type{ 302 clopper_pearson_exact_interval, 303 jeffreys_prior_interval 304 }; 305 306 // 307 // Estimation of the success fraction parameter. 308 // The best estimate is actually simply successes/trials, 309 // these functions are used 310 // to obtain confidence intervals for the success fraction. 311 // find_lower_bound_on_p(RealType trials,RealType successes,RealType probability,interval_type t=clopper_pearson_exact_interval)312 static RealType find_lower_bound_on_p( 313 RealType trials, 314 RealType successes, 315 RealType probability, 316 interval_type t = clopper_pearson_exact_interval) 317 { 318 static const char* function = "boost::math::binomial_distribution<%1%>::find_lower_bound_on_p"; 319 // Error checks: 320 RealType result = 0; 321 if(false == binomial_detail::check_dist_and_k( 322 function, trials, RealType(0), successes, &result, Policy()) 323 && 324 binomial_detail::check_dist_and_prob( 325 function, trials, RealType(0), probability, &result, Policy())) 326 { return result; } 327 328 if(successes == 0) 329 return 0; 330 331 // NOTE!!! The Clopper Pearson formula uses "successes" not 332 // "successes+1" as usual to get the lower bound, 333 // see http://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm 334 return (t == clopper_pearson_exact_interval) ? ibeta_inv(successes, trials - successes + 1, probability, static_cast<RealType*>(0), Policy()) 335 : ibeta_inv(successes + 0.5f, trials - successes + 0.5f, probability, static_cast<RealType*>(0), Policy()); 336 } find_upper_bound_on_p(RealType trials,RealType successes,RealType probability,interval_type t=clopper_pearson_exact_interval)337 static RealType find_upper_bound_on_p( 338 RealType trials, 339 RealType successes, 340 RealType probability, 341 interval_type t = clopper_pearson_exact_interval) 342 { 343 static const char* function = "boost::math::binomial_distribution<%1%>::find_upper_bound_on_p"; 344 // Error checks: 345 RealType result = 0; 346 if(false == binomial_detail::check_dist_and_k( 347 function, trials, RealType(0), successes, &result, Policy()) 348 && 349 binomial_detail::check_dist_and_prob( 350 function, trials, RealType(0), probability, &result, Policy())) 351 { return result; } 352 353 if(trials == successes) 354 return 1; 355 356 return (t == clopper_pearson_exact_interval) ? ibetac_inv(successes + 1, trials - successes, probability, static_cast<RealType*>(0), Policy()) 357 : ibetac_inv(successes + 0.5f, trials - successes + 0.5f, probability, static_cast<RealType*>(0), Policy()); 358 } 359 // Estimate number of trials parameter: 360 // 361 // "How many trials do I need to be P% sure of seeing k events?" 362 // or 363 // "How many trials can I have to be P% sure of seeing fewer than k events?" 364 // find_minimum_number_of_trials(RealType k,RealType p,RealType alpha)365 static RealType find_minimum_number_of_trials( 366 RealType k, // number of events 367 RealType p, // success fraction 368 RealType alpha) // risk level 369 { 370 static const char* function = "boost::math::binomial_distribution<%1%>::find_minimum_number_of_trials"; 371 // Error checks: 372 RealType result = 0; 373 if(false == binomial_detail::check_dist_and_k( 374 function, k, p, k, &result, Policy()) 375 && 376 binomial_detail::check_dist_and_prob( 377 function, k, p, alpha, &result, Policy())) 378 { return result; } 379 380 result = ibetac_invb(k + 1, p, alpha, Policy()); // returns n - k 381 return result + k; 382 } 383 find_maximum_number_of_trials(RealType k,RealType p,RealType alpha)384 static RealType find_maximum_number_of_trials( 385 RealType k, // number of events 386 RealType p, // success fraction 387 RealType alpha) // risk level 388 { 389 static const char* function = "boost::math::binomial_distribution<%1%>::find_maximum_number_of_trials"; 390 // Error checks: 391 RealType result = 0; 392 if(false == binomial_detail::check_dist_and_k( 393 function, k, p, k, &result, Policy()) 394 && 395 binomial_detail::check_dist_and_prob( 396 function, k, p, alpha, &result, Policy())) 397 { return result; } 398 399 result = ibeta_invb(k + 1, p, alpha, Policy()); // returns n - k 400 return result + k; 401 } 402 403 private: 404 RealType m_n; // Not sure if this shouldn't be an int? 405 RealType m_p; // success_fraction 406 }; // template <class RealType, class Policy> class binomial_distribution 407 408 typedef binomial_distribution<> binomial; 409 // typedef binomial_distribution<double> binomial; 410 // IS now included since no longer a name clash with function binomial. 411 //typedef binomial_distribution<double> binomial; // Reserved name of type double. 412 413 template <class RealType, class Policy> range(const binomial_distribution<RealType,Policy> & dist)414 const std::pair<RealType, RealType> range(const binomial_distribution<RealType, Policy>& dist) 415 { // Range of permissible values for random variable k. 416 using boost::math::tools::max_value; 417 return std::pair<RealType, RealType>(static_cast<RealType>(0), dist.trials()); 418 } 419 420 template <class RealType, class Policy> support(const binomial_distribution<RealType,Policy> & dist)421 const std::pair<RealType, RealType> support(const binomial_distribution<RealType, Policy>& dist) 422 { // Range of supported values for random variable k. 423 // This is range where cdf rises from 0 to 1, and outside it, the pdf is zero. 424 return std::pair<RealType, RealType>(static_cast<RealType>(0), dist.trials()); 425 } 426 427 template <class RealType, class Policy> mean(const binomial_distribution<RealType,Policy> & dist)428 inline RealType mean(const binomial_distribution<RealType, Policy>& dist) 429 { // Mean of Binomial distribution = np. 430 return dist.trials() * dist.success_fraction(); 431 } // mean 432 433 template <class RealType, class Policy> variance(const binomial_distribution<RealType,Policy> & dist)434 inline RealType variance(const binomial_distribution<RealType, Policy>& dist) 435 { // Variance of Binomial distribution = np(1-p). 436 return dist.trials() * dist.success_fraction() * (1 - dist.success_fraction()); 437 } // variance 438 439 template <class RealType, class Policy> 440 RealType pdf(const binomial_distribution<RealType, Policy>& dist, const RealType& k) 441 { // Probability Density/Mass Function. 442 BOOST_FPU_EXCEPTION_GUARD 443 444 BOOST_MATH_STD_USING // for ADL of std functions 445 446 RealType n = dist.trials(); 447 448 // Error check: 449 RealType result = 0; // initialization silences some compiler warnings 450 if(false == binomial_detail::check_dist_and_k( 451 "boost::math::pdf(binomial_distribution<%1%> const&, %1%)", 452 n, 453 dist.success_fraction(), 454 k, 455 &result, Policy())) 456 { 457 return result; 458 } 459 460 // Special cases of success_fraction, regardless of k successes and regardless of n trials. 461 if (dist.success_fraction() == 0) 462 { // probability of zero successes is 1: 463 return static_cast<RealType>(k == 0 ? 1 : 0); 464 } 465 if (dist.success_fraction() == 1) 466 { // probability of n successes is 1: 467 return static_cast<RealType>(k == n ? 1 : 0); 468 } 469 // k argument may be integral, signed, or unsigned, or floating point. 470 // If necessary, it has already been promoted from an integral type. 471 if (n == 0) 472 { 473 return 1; // Probability = 1 = certainty. 474 } 475 if (k == 0) 476 { // binomial coeffic (n 0) = 1, 477 // n ^ 0 = 1 478 return pow(1 - dist.success_fraction(), n); 479 } 480 if (k == n) 481 { // binomial coeffic (n n) = 1, 482 // n ^ 0 = 1 483 return pow(dist.success_fraction(), k); // * pow((1 - dist.success_fraction()), (n - k)) = 1 484 } 485 486 // Probability of getting exactly k successes 487 // if C(n, k) is the binomial coefficient then: 488 // 489 // f(k; n,p) = C(n, k) * p^k * (1-p)^(n-k) 490 // = (n!/(k!(n-k)!)) * p^k * (1-p)^(n-k) 491 // = (tgamma(n+1) / (tgamma(k+1)*tgamma(n-k+1))) * p^k * (1-p)^(n-k) 492 // = p^k (1-p)^(n-k) / (beta(k+1, n-k+1) * (n+1)) 493 // = ibeta_derivative(k+1, n-k+1, p) / (n+1) 494 // 495 using boost::math::ibeta_derivative; // a, b, x 496 return ibeta_derivative(k+1, n-k+1, dist.success_fraction(), Policy()) / (n+1); 497 498 } // pdf 499 500 template <class RealType, class Policy> cdf(const binomial_distribution<RealType,Policy> & dist,const RealType & k)501 inline RealType cdf(const binomial_distribution<RealType, Policy>& dist, const RealType& k) 502 { // Cumulative Distribution Function Binomial. 503 // The random variate k is the number of successes in n trials. 504 // k argument may be integral, signed, or unsigned, or floating point. 505 // If necessary, it has already been promoted from an integral type. 506 507 // Returns the sum of the terms 0 through k of the Binomial Probability Density/Mass: 508 // 509 // i=k 510 // -- ( n ) i n-i 511 // > | | p (1-p) 512 // -- ( i ) 513 // i=0 514 515 // The terms are not summed directly instead 516 // the incomplete beta integral is employed, 517 // according to the formula: 518 // P = I[1-p]( n-k, k+1). 519 // = 1 - I[p](k + 1, n - k) 520 521 BOOST_MATH_STD_USING // for ADL of std functions 522 523 RealType n = dist.trials(); 524 RealType p = dist.success_fraction(); 525 526 // Error check: 527 RealType result = 0; 528 if(false == binomial_detail::check_dist_and_k( 529 "boost::math::cdf(binomial_distribution<%1%> const&, %1%)", 530 n, 531 p, 532 k, 533 &result, Policy())) 534 { 535 return result; 536 } 537 if (k == n) 538 { 539 return 1; 540 } 541 542 // Special cases, regardless of k. 543 if (p == 0) 544 { // This need explanation: 545 // the pdf is zero for all cases except when k == 0. 546 // For zero p the probability of zero successes is one. 547 // Therefore the cdf is always 1: 548 // the probability of k or *fewer* successes is always 1 549 // if there are never any successes! 550 return 1; 551 } 552 if (p == 1) 553 { // This is correct but needs explanation: 554 // when k = 1 555 // all the cdf and pdf values are zero *except* when k == n, 556 // and that case has been handled above already. 557 return 0; 558 } 559 // 560 // P = I[1-p](n - k, k + 1) 561 // = 1 - I[p](k + 1, n - k) 562 // Use of ibetac here prevents cancellation errors in calculating 563 // 1-p if p is very small, perhaps smaller than machine epsilon. 564 // 565 // Note that we do not use a finite sum here, since the incomplete 566 // beta uses a finite sum internally for integer arguments, so 567 // we'll just let it take care of the necessary logic. 568 // 569 return ibetac(k + 1, n - k, p, Policy()); 570 } // binomial cdf 571 572 template <class RealType, class Policy> cdf(const complemented2_type<binomial_distribution<RealType,Policy>,RealType> & c)573 inline RealType cdf(const complemented2_type<binomial_distribution<RealType, Policy>, RealType>& c) 574 { // Complemented Cumulative Distribution Function Binomial. 575 // The random variate k is the number of successes in n trials. 576 // k argument may be integral, signed, or unsigned, or floating point. 577 // If necessary, it has already been promoted from an integral type. 578 579 // Returns the sum of the terms k+1 through n of the Binomial Probability Density/Mass: 580 // 581 // i=n 582 // -- ( n ) i n-i 583 // > | | p (1-p) 584 // -- ( i ) 585 // i=k+1 586 587 // The terms are not summed directly instead 588 // the incomplete beta integral is employed, 589 // according to the formula: 590 // Q = 1 -I[1-p]( n-k, k+1). 591 // = I[p](k + 1, n - k) 592 593 BOOST_MATH_STD_USING // for ADL of std functions 594 595 RealType const& k = c.param; 596 binomial_distribution<RealType, Policy> const& dist = c.dist; 597 RealType n = dist.trials(); 598 RealType p = dist.success_fraction(); 599 600 // Error checks: 601 RealType result = 0; 602 if(false == binomial_detail::check_dist_and_k( 603 "boost::math::cdf(binomial_distribution<%1%> const&, %1%)", 604 n, 605 p, 606 k, 607 &result, Policy())) 608 { 609 return result; 610 } 611 612 if (k == n) 613 { // Probability of greater than n successes is necessarily zero: 614 return 0; 615 } 616 617 // Special cases, regardless of k. 618 if (p == 0) 619 { 620 // This need explanation: the pdf is zero for all 621 // cases except when k == 0. For zero p the probability 622 // of zero successes is one. Therefore the cdf is always 623 // 1: the probability of *more than* k successes is always 0 624 // if there are never any successes! 625 return 0; 626 } 627 if (p == 1) 628 { 629 // This needs explanation, when p = 1 630 // we always have n successes, so the probability 631 // of more than k successes is 1 as long as k < n. 632 // The k == n case has already been handled above. 633 return 1; 634 } 635 // 636 // Calculate cdf binomial using the incomplete beta function. 637 // Q = 1 -I[1-p](n - k, k + 1) 638 // = I[p](k + 1, n - k) 639 // Use of ibeta here prevents cancellation errors in calculating 640 // 1-p if p is very small, perhaps smaller than machine epsilon. 641 // 642 // Note that we do not use a finite sum here, since the incomplete 643 // beta uses a finite sum internally for integer arguments, so 644 // we'll just let it take care of the necessary logic. 645 // 646 return ibeta(k + 1, n - k, p, Policy()); 647 } // binomial cdf 648 649 template <class RealType, class Policy> quantile(const binomial_distribution<RealType,Policy> & dist,const RealType & p)650 inline RealType quantile(const binomial_distribution<RealType, Policy>& dist, const RealType& p) 651 { 652 return binomial_detail::quantile_imp(dist, p, RealType(1-p)); 653 } // quantile 654 655 template <class RealType, class Policy> 656 RealType quantile(const complemented2_type<binomial_distribution<RealType, Policy>, RealType>& c) 657 { 658 return binomial_detail::quantile_imp(c.dist, RealType(1-c.param), c.param); 659 } // quantile 660 661 template <class RealType, class Policy> mode(const binomial_distribution<RealType,Policy> & dist)662 inline RealType mode(const binomial_distribution<RealType, Policy>& dist) 663 { 664 BOOST_MATH_STD_USING // ADL of std functions. 665 RealType p = dist.success_fraction(); 666 RealType n = dist.trials(); 667 return floor(p * (n + 1)); 668 } 669 670 template <class RealType, class Policy> median(const binomial_distribution<RealType,Policy> & dist)671 inline RealType median(const binomial_distribution<RealType, Policy>& dist) 672 { // Bounds for the median of the negative binomial distribution 673 // VAN DE VEN R. ; WEBER N. C. ; 674 // Univ. Sydney, school mathematics statistics, Sydney N.S.W. 2006, AUSTRALIE 675 // Metrika (Metrika) ISSN 0026-1335 CODEN MTRKA8 676 // 1993, vol. 40, no3-4, pp. 185-189 (4 ref.) 677 678 // Bounds for median and 50 percetage point of binomial and negative binomial distribution 679 // Metrika, ISSN 0026-1335 (Print) 1435-926X (Online) 680 // Volume 41, Number 1 / December, 1994, DOI 10.1007/BF01895303 681 BOOST_MATH_STD_USING // ADL of std functions. 682 RealType p = dist.success_fraction(); 683 RealType n = dist.trials(); 684 // Wikipedia says one of floor(np) -1, floor (np), floor(np) +1 685 return floor(p * n); // Chose the middle value. 686 } 687 688 template <class RealType, class Policy> skewness(const binomial_distribution<RealType,Policy> & dist)689 inline RealType skewness(const binomial_distribution<RealType, Policy>& dist) 690 { 691 BOOST_MATH_STD_USING // ADL of std functions. 692 RealType p = dist.success_fraction(); 693 RealType n = dist.trials(); 694 return (1 - 2 * p) / sqrt(n * p * (1 - p)); 695 } 696 697 template <class RealType, class Policy> kurtosis(const binomial_distribution<RealType,Policy> & dist)698 inline RealType kurtosis(const binomial_distribution<RealType, Policy>& dist) 699 { 700 RealType p = dist.success_fraction(); 701 RealType n = dist.trials(); 702 return 3 - 6 / n + 1 / (n * p * (1 - p)); 703 } 704 705 template <class RealType, class Policy> kurtosis_excess(const binomial_distribution<RealType,Policy> & dist)706 inline RealType kurtosis_excess(const binomial_distribution<RealType, Policy>& dist) 707 { 708 RealType p = dist.success_fraction(); 709 RealType q = 1 - p; 710 RealType n = dist.trials(); 711 return (1 - 6 * p * q) / (n * p * q); 712 } 713 714 } // namespace math 715 } // namespace boost 716 717 // This include must be at the end, *after* the accessors 718 // for this distribution have been defined, in order to 719 // keep compilers that support two-phase lookup happy. 720 #include <boost/math/distributions/detail/derived_accessors.hpp> 721 722 #endif // BOOST_MATH_SPECIAL_BINOMIAL_HPP 723 724 725