1[section:triangular_dist Triangular Distribution] 2 3 4``#include <boost/math/distributions/triangular.hpp>`` 5 6 namespace boost{ namespace math{ 7 template <class RealType = double, 8 class ``__Policy`` = ``__policy_class`` > 9 class triangular_distribution; 10 11 typedef triangular_distribution<> triangular; 12 13 template <class RealType, class ``__Policy``> 14 class triangular_distribution 15 { 16 public: 17 typedef RealType value_type; 18 typedef Policy policy_type; 19 20 triangular_distribution(RealType lower = -1, RealType mode = 0) RealType upper = 1); // Constructor. 21 : m_lower(lower), m_mode(mode), m_upper(upper) // Default is -1, 0, +1 symmetric triangular distribution. 22 // Accessor functions. 23 RealType lower()const; 24 RealType mode()const; 25 RealType upper()const; 26 }; // class triangular_distribution 27 28 }} // namespaces 29 30The [@http://en.wikipedia.org/wiki/Triangular_distribution triangular distribution] 31is a [@http://en.wikipedia.org/wiki/Continuous_distribution continuous] 32[@http://en.wikipedia.org/wiki/Probability_distribution probability distribution] 33with a lower limit a, 34[@http://en.wikipedia.org/wiki/Mode_%28statistics%29 mode c], 35and upper limit b. 36 37The triangular distribution is often used where the distribution is only vaguely known, 38but, like the [@http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 uniform distribution], 39upper and limits are 'known', but a 'best guess', the mode or center point, is also added. 40It has been recommended as a 41[@http://www.worldscibooks.com/mathematics/etextbook/5720/5720_chap1.pdf proxy for the beta distribution.] 42The distribution is used in business decision making and project planning. 43 44The [@http://en.wikipedia.org/wiki/Triangular_distribution triangular distribution] 45is a distribution with the 46[@http://en.wikipedia.org/wiki/Probability_density_function probability density function]: 47 48[expression f(x) = 2(x-a)/(b-a) (c-a) [sixemspace] for a <= x <= c] 49[expression f(x) = 2(b-x)/(b-a) (b-c) [sixemspace] for c < x <= b] 50 51Parameter ['a] (lower) can be any finite value. 52Parameter ['b] (upper) can be any finite value > a (lower). 53Parameter ['c] (mode) a <= c <= b. This is the most probable value. 54 55The [@http://en.wikipedia.org/wiki/Random_variate random variate] x must also be finite, and is supported lower <= x <= upper. 56 57The triangular distribution may be appropriate when an assumption of a normal distribution 58is unjustified because uncertainty is caused by rounding and quantization from analog to digital conversion. 59Upper and lower limits are known, and the most probable value lies midway. 60 61The distribution simplifies when the 'best guess' is either the lower or upper limit - a 90 degree angle triangle. 62The 001 triangular distribution which expresses an estimate that the lowest value is the most likely; 63for example, you believe that the next-day quoted delivery date is most likely 64(knowing that a quicker delivery is impossible - the postman only comes once a day), 65and that longer delays are decreasingly likely, 66and delivery is assumed to never take more than your upper limit. 67 68The following graph illustrates how the 69[@http://en.wikipedia.org/wiki/Probability_density_function probability density function PDF] 70varies with the various parameters: 71 72[graph triangular_pdf] 73 74and cumulative distribution function 75 76[graph triangular_cdf] 77 78[h4 Member Functions] 79 80 triangular_distribution(RealType lower = 0, RealType mode = 0 RealType upper = 1); 81 82Constructs a [@http://en.wikipedia.org/wiki/triangular_distribution triangular distribution] 83with lower /lower/ (a) and upper /upper/ (b). 84 85Requires that the /lower/, /mode/ and /upper/ parameters are all finite, 86otherwise calls __domain_error. 87 88[warning These constructors are slightly different from the analogs provided by __Mathworld 89[@http://reference.wolfram.com/language/ref/TriangularDistribution.html Triangular distribution], 90where 91 92[^TriangularDistribution\[{min, max}\]] represents a [*symmetric] triangular statistical distribution giving values between min and max.[br] 93[^TriangularDistribution\[\]] represents a [*symmetric] triangular statistical distribution giving values between 0 and 1.[br] 94[^TriangularDistribution\[{min, max}, c\]] represents a triangular distribution with mode at c (usually [*asymmetric]).[br] 95 96So, for example, to compute a variance using __WolframAlpha, use 97[^N\[variance\[TriangularDistribution{1, +2}\], 50\]] 98] 99 100The parameters of a distribution can be obtained using these member functions: 101 102 RealType lower()const; 103 104Returns the ['lower] parameter of this distribution (default -1). 105 106 RealType mode()const; 107 108Returns the ['mode] parameter of this distribution (default 0). 109 110 RealType upper()const; 111 112Returns the ['upper] parameter of this distribution (default+1). 113 114[h4 Non-member Accessors] 115 116All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all 117distributions are supported: __usual_accessors. 118 119The domain of the random variable is \lower\ to \upper\, 120and the supported range is lower <= x <= upper. 121 122[h4 Accuracy] 123 124The triangular distribution is implemented with simple arithmetic operators and so should have errors within an epsilon or two, 125except quantiles with arguments nearing the extremes of zero and unity. 126 127[h4 Implementation] 128 129In the following table, a is the /lower/ parameter of the distribution, 130c is the /mode/ parameter, 131b is the /upper/ parameter, 132/x/ is the random variate, /p/ is the probability and /q = 1-p/. 133 134[table 135[[Function][Implementation Notes]] 136[[pdf][Using the relation: pdf = 0 for x < mode, 2(x-a)\/(b-a)(c-a) else 2*(b-x)\/((b-a)(b-c))]] 137[[cdf][Using the relation: cdf = 0 for x < mode (x-a)[super 2]\/((b-a)(c-a)) else 1 - (b-x)[super 2]\/((b-a)(b-c))]] 138[[cdf complement][Using the relation: q = 1 - p ]] 139[[quantile][let p0 = (c-a)\/(b-a) the point of inflection on the cdf, 140then given probability p and q = 1-p: 141 142x = sqrt((b-a)(c-a)p) + a ; for p < p0 143 144x = c ; for p == p0 145 146x = b - sqrt((b-a)(b-c)q) ; for p > p0 147 148(See [@../../../../boost/math/distributions/triangular.hpp /boost/math/distributions/triangular.hpp] for details.)]] 149[[quantile from the complement][As quantile (See [@../../../../boost/math/distributions/triangular.hpp /boost/math/distributions/triangular.hpp] for details.)]] 150[[mean][(a + b + 3) \/ 3 ]] 151[[variance][(a[super 2]+b[super 2]+c[super 2] - ab - ac - bc)\/18]] 152[[mode][c]] 153[[skewness][(See [@../../../../boost/math/distributions/triangular.hpp /boost/math/distributions/triangular.hpp] for details). ]] 154[[kurtosis][12\/5]] 155[[kurtosis excess][-3\/5]] 156] 157 158Some 'known good' test values were obtained using __WolframAlpha. 159 160[h4 References] 161 162* [@http://en.wikipedia.org/wiki/Triangular_distribution Wikipedia triangular distribution] 163* [@http://mathworld.wolfram.com/TriangularDistribution.html Weisstein, Eric W. "Triangular Distribution." From MathWorld--A Wolfram Web Resource.] 164* Evans, M.; Hastings, N.; and Peacock, B. "Triangular Distribution." Ch. 40 in Statistical Distributions, 3rd ed. New York: Wiley, pp. 187-188, 2000, ISBN - 0471371246. 165* [@http://www.measurement.sk/2002/S1/Wimmer2.pdf Gejza Wimmer, Viktor Witkovsky and Tomas Duby, 166Measurement Science Review, Volume 2, Section 1, 2002, Proper Rounding Of The Measurement Results Under The Assumption Of Triangular Distribution.] 167 168[endsect][/section:triangular_dist triangular] 169 170[/ 171 Copyright 2006 John Maddock and Paul A. Bristow. 172 Distributed under the Boost Software License, Version 1.0. 173 (See accompanying file LICENSE_1_0.txt or copy at 174 http://www.boost.org/LICENSE_1_0.txt). 175] 176 177