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289 lines
11 KiB
Plaintext
289 lines
11 KiB
Plaintext
[section:arcine_dist Arcsine Distribution]
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[import ../../example/arcsine_example.cpp] [/ for arcsine snips below]
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``#include <boost/math/distributions/arcsine.hpp>``
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namespace boost{ namespace math{
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template <class RealType = double,
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class ``__Policy`` = ``__policy_class`` >
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class arcsine_distribution;
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typedef arcsine_distribution<double> arcsine; // double precision standard arcsine distribution [0,1].
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template <class RealType, class ``__Policy``>
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class arcsine_distribution
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{
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public:
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typedef RealType value_type;
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typedef Policy policy_type;
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// Constructor from two range parameters, x_min and x_max:
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arcsine_distribution(RealType x_min, RealType x_max);
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// Range Parameter accessors:
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RealType x_min() const;
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RealType x_max() const;
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};
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}} // namespaces
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The class type `arcsine_distribution` represents an
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[@http://en.wikipedia.org/wiki/arcsine_distribution arcsine]
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[@http://en.wikipedia.org/wiki/Probability_distribution probability distribution function].
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The arcsine distribution is named because its CDF uses the inverse sin[super -1] or arcsine.
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This is implemented as a generalized version with support from ['x_min] to ['x_max]
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providing the 'standard arcsine distribution' as default with ['x_min = 0] and ['x_max = 1].
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(A few make other choices for 'standard').
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The arcsine distribution is generalized to include any bounded support ['a <= x <= b] by
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[@http://reference.wolfram.com/language/ref/ArcSinDistribution.html Wolfram] and
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[@http://en.wikipedia.org/wiki/arcsine_distribution Wikipedia],
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but also using ['location] and ['scale] parameters by
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[@http://www.math.uah.edu/stat/index.html Virtual Laboratories in Probability and Statistics]
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[@http://www.math.uah.edu/stat/special/Arcsine.html Arcsine distribution].
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The end-point version is simpler and more obvious, so we implement that.
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If desired, [@http://en.wikipedia.org/wiki/arcsine_distribution this]
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outlines how the __beta_distrib can be used to add a shape factor.
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The [@http://en.wikipedia.org/wiki/Probability_density_function probability density function PDF]
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for the [@http://en.wikipedia.org/wiki/arcsine_distribution arcsine distribution]
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defined on the interval \[['x_min, x_max]\] is given by:
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[figspace] [figspace] f(x; x_min, x_max) = 1 /([pi][sdot][sqrt]((x - x_min)[sdot](x_max - x_min))
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For example, __WolframAlpha arcsine distribution, from input of
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N[PDF[arcsinedistribution[0, 1], 0.5], 50]
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computes the PDF value
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0.63661977236758134307553505349005744813783858296183
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The Probability Density Functions (PDF) of generalized arcsine distributions are symmetric U-shaped curves,
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centered on ['(x_max - x_min)/2],
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highest (infinite) near the two extrema, and quite flat over the central region.
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If random variate ['x] is ['x_min] or ['x_max], then the PDF is infinity.
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If random variate ['x] is ['x_min] then the CDF is zero.
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If random variate ['x] is ['x_max] then the CDF is unity.
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The 'Standard' (0, 1) arcsine distribution is shown in blue
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and some generalized examples with other ['x] ranges.
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[graph arcsine_pdf]
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The Cumulative Distribution Function CDF is defined as
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[figspace] [figspace] F(x) = 2[sdot]arcsin([sqrt]((x-x_min)/(x_max - x))) / [pi]
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[graph arcsine_cdf]
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[h5 Constructor]
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arcsine_distribution(RealType x_min, RealType x_max);
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constructs an arcsine distribution with range parameters ['x_min] and ['x_max].
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Requires ['x_min < x_max], otherwise __domain_error is called.
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For example:
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arcsine_distribution<> myarcsine(-2, 4);
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constructs an arcsine distribution with ['x_min = -2] and ['x_max = 4].
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Default values of ['x_min = 0] and ['x_max = 1] and a ` typedef arcsine_distribution<double> arcsine;` mean that
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arcsine as;
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constructs a 'Standard 01' arcsine distribution.
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[h5 Parameter Accessors]
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RealType x_min() const;
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RealType x_max() const;
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Return the parameter ['x_min] or ['x_max] from which this distribution was constructed.
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So, for example:
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[arcsine_snip_8]
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[h4 Non-member Accessor Functions]
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All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions]
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that are generic to all distributions are supported: __usual_accessors.
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The formulae for calculating these are shown in the table below, and at
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[@http://mathworld.wolfram.com/arcsineDistribution.html Wolfram Mathworld].
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[note There are always [*two] values for the [*mode], at ['x_min] and at ['x_max], default 0 and 1,
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so instead we raise the exception __domain_error.
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At these extrema, the PDFs are infinite, and the CDFs zero or unity.]
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[h4 Applications]
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The arcsine distribution is useful to describe
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[@http://en.wikipedia.org/wiki/Random_walk Random walks], (including drunken walks)
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[@http://en.wikipedia.org/wiki/Brownian_motion Brownian motion],
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[@http://en.wikipedia.org/wiki/Wiener_process Weiner processes],
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[@http://en.wikipedia.org/wiki/Bernoulli_trial Bernoulli trials],
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and their appplication to solve stock market and other
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[@http://en.wikipedia.org/wiki/Gambler%27s_ruin ruinous gambling games].
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The random variate ['x] is constrained to ['x_min] and ['x_max], (for our 'standard' distribution, 0 and 1),
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and is usually some fraction. For any other ['x_min] and ['x_max] a fraction can be obtained from ['x] using
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[sixemspace] fraction = (x - x_min) / (x_max - x_min)
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The simplest example is tossing heads and tails with a fair coin and modelling the risk of losing, or winning.
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Walkers (molecules, drunks...) moving left or right of a centre line are another common example.
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The random variate ['x] is the fraction of time spent on the 'winning' side.
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If half the time is spent on the 'winning' side (and so the other half on the 'losing' side) then ['x = 1/2].
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For large numbers of tosses, this is modelled by the (standard \[0,1\]) arcsine distribution,
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and the PDF can be calculated thus:
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[arcsine_snip_2]
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From the plot of PDF, it is clear that ['x] = [frac12] is the [*minimum] of the curve,
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so this is the [*least likely] scenario.
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(This is highly counter-intuitive, considering that fair tosses must [*eventually] become equal.
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It turns out that ['eventually] is not just very long, but [*infinite]!).
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The [*most likely] scenarios are towards the extrema where ['x] = 0 or ['x] = 1.
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If fraction of time on the left is a [frac14],
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it is only slightly more likely because the curve is quite flat bottomed.
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[arcsine_snip_3]
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If we consider fair coin-tossing games being played for 100 days
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(hypothetically continuously to be 'at-limit')
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the person winning after day 5 will not change in fraction 0.144 of the cases.
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We can easily compute this setting ['x] = 5./100 = 0.05
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[arcsine_snip_4]
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Similarly, we can compute from a fraction of 0.05 /2 = 0.025
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(halved because we are considering both winners and losers)
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corresponding to 1 - 0.025 or 97.5% of the gamblers, (walkers, particles...) on the [*same side] of the origin
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[arcsine_snip_5]
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(use of the complement gives a bit more clarity,
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and avoids potential loss of accuracy when ['x] is close to unity, see __why_complements).
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[arcsine_snip_6]
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or we can reverse the calculation by assuming a fraction of time on one side, say fraction 0.2,
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[arcsine_snip_7]
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[*Summary]: Every time we toss, the odds are equal,
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so on average we have the same change of winning and losing.
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But this is [*not true] for an an individual game where one will be [*mostly in a bad or good patch].
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This is quite counter-intuitive to most people, but the mathematics is clear,
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and gamblers continue to provide proof.
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[*Moral]: if you in a losing patch, leave the game.
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(Because the odds to recover to a good patch are poor).
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[*Corollary]: Quit while you are ahead?
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A working example is at [@../../example/arcsine_example.cpp arcsine_example.cpp]
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including sample output .
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[h4 Related distributions]
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The arcsine distribution with ['x_min = 0] and ['x_max = 1] is special case of the
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__beta_distrib with [alpha] = 1/2 and [beta] = 1/2.
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[h4 Accuracy]
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This distribution is implemented using sqrt, sine, cos and arc sine and cos trigonometric functions
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which are normally accurate to a few __epsilon.
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But all values suffer from [@http://en.wikipedia.org/wiki/Loss_of_significance loss of significance or cancellation error]
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for values of ['x] close to ['x_max].
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For example, for a standard [0, 1] arcsine distribution ['as], the pdf is symmetric about random variate ['x = 0.5]
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so that one would expect `pdf(as, 0.01) == pdf(as, 0.99)`. But as ['x] nears unity, there is increasing
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[@http://en.wikipedia.org/wiki/Loss_of_significance loss of significance].
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To counteract this, the complement versions of CDF and quantile
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are implemented with alternative expressions using ['cos[super -1]] instead of ['sin[super -1]].
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Users should see __why_complements for guidance on when to avoid loss of accuracy by using complements.
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[h4 Testing]
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The results were tested against a few accurate spot values computed by __WolframAlpha, for example:
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N[PDF[arcsinedistribution[0, 1], 0.5], 50]
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0.63661977236758134307553505349005744813783858296183
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[h4 Implementation]
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In the following table ['a] and ['b] are the parameters ['x_min][space] and ['x_max],
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['x] is the random variable, ['p] is the probability and its complement ['q = 1-p].
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[table
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[[Function][Implementation Notes]]
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[[support] [x [isin] \[a, b\], default x [isin] \[0, 1\] ]]
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[[pdf] [f(x; a, b) = 1/([pi][sdot][sqrt](x - a)[sdot](b - x))]]
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[[cdf] [F(x) = 2/[pi][sdot]sin[super-1]([sqrt](x - a) / (b - a) ) ]]
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[[cdf of complement] [2/([pi][sdot]cos[super-1]([sqrt](x - a) / (b - a)))]]
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[[quantile] [-a[sdot]sin[super 2]([frac12][pi][sdot]p) + a + b[sdot]sin[super 2]([frac12][pi][sdot]p)]]
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[[quantile from the complement] [-a[sdot]cos[super 2]([frac12][pi][sdot]p) + a + b[sdot]cos[super 2]([frac12][pi][sdot]q)]]
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[[mean] [[frac12](a+b)]]
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[[median] [[frac12](a+b)]]
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[[mode] [ x [isin] \[a, b\], so raises domain_error (returning NaN).]]
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[[variance] [(b - a)[super 2] / 8]]
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[[skewness] [0]]
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[[kurtosis excess] [ -3/2 ]]
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[[kurtosis] [kurtosis_excess + 3]]
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]
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The quantile was calculated using an expression obtained by using __WolframAlpha
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to invert the formula for the CDF thus
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solve [p - 2/pi sin^-1(sqrt((x-a)/(b-a))) = 0, x]
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which was interpreted as
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Solve[p - (2 ArcSin[Sqrt[(-a + x)/(-a + b)]])/Pi == 0, x, MaxExtraConditions -> Automatic]
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and produced the resulting expression
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x = -a sin^2((pi p)/2)+a+b sin^2((pi p)/2)
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Thanks to Wolfram for providing this facility.
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[h4 References]
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* [@http://en.wikipedia.org/wiki/arcsine_distribution Wikipedia arcsine distribution]
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* [@http://en.wikipedia.org/wiki/Beta_distribution Wikipedia Beta distribution]
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* [@http://mathworld.wolfram.com/BetaDistribution.html Wolfram MathWorld]
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* [@http://www.wolframalpha.com/ Wolfram Alpha]
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[h4 Sources]
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*[@http://estebanmoro.org/2009/04/the-probability-of-going-through-a-bad-patch The probability of going through a bad patch] Esteban Moro's Blog.
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*[@http://www.gotohaggstrom.com/What%20do%20schmucks%20and%20the%20arc%20sine%20law%20have%20in%20common.pdf What soschumcks and the arc sine have in common] Peter Haggstrom.
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*[@http://www.math.uah.edu/stat/special/Arcsine.html arcsine distribution].
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*[@http://reference.wolfram.com/language/ref/ArcSinDistribution.html Wolfram reference arcsine examples].
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*[@http://www.math.harvard.edu/library/sternberg/slides/1180908.pdf Shlomo Sternberg slides].
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[endsect] [/section:arcsine_dist arcsine]
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[/ arcsine.qbk
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Copyright 2014 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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