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  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.arcine_dist"></a><a class="link" href="arcine_dist.html" title="Arcsine Distribution">Arcsine Distribution</a>
  28 </h4></div></div></div>
  29 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">arcsine</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
  30 <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
  31
  32  <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
  33            <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a>   <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
  34 <span class="keyword">class</span> <span class="identifier">arcsine_distribution</span><span class="special">;</span>
  35
  36 <span class="keyword">typedef</span> <span class="identifier">arcsine_distribution</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">arcsine</span><span class="special">;</span> <span class="comment">// double precision standard arcsine distribution [0,1].</span>
  37
  38 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  39 <span class="keyword">class</span> <span class="identifier">arcsine_distribution</span>
  40 <span class="special">{</span>
  41 <span class="keyword">public</span><span class="special">:</span>
  42    <span class="keyword">typedef</span> <span class="identifier">RealType</span>  <span class="identifier">value_type</span><span class="special">;</span>
  43    <span class="keyword">typedef</span> <span class="identifier">Policy</span>    <span class="identifier">policy_type</span><span class="special">;</span>
  44
  45    <span class="comment">// Constructor from two range parameters, x_min and x_max:</span>
  46    <span class="identifier">arcsine_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">x_min</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">x_max</span><span class="special">);</span>
  47
  48    <span class="comment">// Range Parameter accessors:</span>
  49    <span class="identifier">RealType</span> <span class="identifier">x_min</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
  50    <span class="identifier">RealType</span> <span class="identifier">x_max</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
  51 <span class="special">};</span>
  52 <span class="special">}}</span> <span class="comment">// namespaces</span>
  53 </pre>
  54 <p>
  55           The class type <code class="computeroutput"><span class="identifier">arcsine_distribution</span></code>
  56           represents an <a href="http://en.wikipedia.org/wiki/arcsine_distribution" target="_top">arcsine</a>
  57           <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">probability
  58           distribution function</a>. The arcsine distribution is named because
  59           its CDF uses the inverse sin<sup>-1</sup> or arcsine.
  60         </p>
  61 <p>
  62           This is implemented as a generalized version with support from <span class="emphasis"><em>x_min</em></span>
  63           to <span class="emphasis"><em>x_max</em></span> providing the 'standard arcsine distribution'
  64           as default with <span class="emphasis"><em>x_min = 0</em></span> and <span class="emphasis"><em>x_max = 1</em></span>.
  65           (A few make other choices for 'standard').
  66         </p>
  67 <p>
  68           The arcsine distribution is generalized to include any bounded support
  69           <span class="emphasis"><em>a &lt;= x &lt;= b</em></span> by <a href="http://reference.wolfram.com/language/ref/ArcSinDistribution.html" target="_top">Wolfram</a>
  70           and <a href="http://en.wikipedia.org/wiki/arcsine_distribution" target="_top">Wikipedia</a>,
  71           but also using <span class="emphasis"><em>location</em></span> and <span class="emphasis"><em>scale</em></span>
  72           parameters by <a href="http://www.math.uah.edu/stat/index.html" target="_top">Virtual
  73           Laboratories in Probability and Statistics</a> <a href="http://www.math.uah.edu/stat/special/Arcsine.html" target="_top">Arcsine
  74           distribution</a>. The end-point version is simpler and more obvious,
  75           so we implement that. If desired, <a href="http://en.wikipedia.org/wiki/arcsine_distribution" target="_top">this</a>
  76           outlines how the <a class="link" href="beta_dist.html" title="Beta Distribution">Beta
  77           Distribution</a> can be used to add a shape factor.
  78         </p>
  79 <p>
  80           The <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
  81           density function PDF</a> for the <a href="http://en.wikipedia.org/wiki/arcsine_distribution" target="_top">arcsine
  82           distribution</a> defined on the interval [<span class="emphasis"><em>x_min, x_max</em></span>]
  83           is given by:
  84         </p>
  85 <div class="blockquote"><blockquote class="blockquote"><p>
  86             <span class="serif_italic">f(x; x_min, x_max) = 1 /(&#960;&#8901;&#8730;((x - x_min)&#8901;(x_max
  87             - x_min))</span>
  88           </p></blockquote></div>
  89 <p>
  90           For example, <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a>
  91           arcsine distribution, from input of
  92         </p>
  93 <pre class="programlisting"><span class="identifier">N</span><span class="special">[</span><span class="identifier">PDF</span><span class="special">[</span><span class="identifier">arcsinedistribution</span><span class="special">[</span><span class="number">0</span><span class="special">,</span> <span class="number">1</span><span class="special">],</span> <span class="number">0.5</span><span class="special">],</span> <span class="number">50</span><span class="special">]</span>
  94 </pre>
  95 <p>
  96           computes the PDF value
  97         </p>
  98 <pre class="programlisting"><span class="number">0.63661977236758134307553505349005744813783858296183</span>
  99 </pre>
 100 <p>
 101           The Probability Density Functions (PDF) of generalized arcsine distributions
 102           are symmetric U-shaped curves, centered on <span class="emphasis"><em>(x_max - x_min)/2</em></span>,
 103           highest (infinite) near the two extrema, and quite flat over the central
 104           region.
 105         </p>
 106 <p>
 107           If random variate <span class="emphasis"><em>x</em></span> is <span class="emphasis"><em>x_min</em></span>
 108           or <span class="emphasis"><em>x_max</em></span>, then the PDF is infinity. If random variate
 109           <span class="emphasis"><em>x</em></span> is <span class="emphasis"><em>x_min</em></span> then the CDF is zero.
 110           If random variate <span class="emphasis"><em>x</em></span> is <span class="emphasis"><em>x_max</em></span>
 111           then the CDF is unity.
 112         </p>
 113 <p>
 114           The 'Standard' (0, 1) arcsine distribution is shown in blue and some generalized
 115           examples with other <span class="emphasis"><em>x</em></span> ranges.
 116         </p>
 117 <div class="blockquote"><blockquote class="blockquote"><p>
 118             <span class="inlinemediaobject"><img src="../../../../graphs/arcsine_pdf.svg" align="middle"></span>
 119
 120           </p></blockquote></div>
 121 <p>
 122           The Cumulative Distribution Function CDF is defined as
 123         </p>
 124 <div class="blockquote"><blockquote class="blockquote"><p>
 125             <span class="serif_italic">F(x) = 2&#8901;arcsin(&#8730;((x-x_min)/(x_max - x))) /
 126             &#960;</span>
 127           </p></blockquote></div>
 128 <div class="blockquote"><blockquote class="blockquote"><p>
 129             <span class="inlinemediaobject"><img src="../../../../graphs/arcsine_cdf.svg" align="middle"></span>
 130
 131           </p></blockquote></div>
 132 <h6>
 133 <a name="math_toolkit.dist_ref.dists.arcine_dist.h0"></a>
 134           <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.constructor"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.constructor">Constructor</a>
 135         </h6>
 136 <pre class="programlisting"><span class="identifier">arcsine_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">x_min</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">x_max</span><span class="special">);</span>
 137 </pre>
 138 <p>
 139           constructs an arcsine distribution with range parameters <span class="emphasis"><em>x_min</em></span>
 140           and <span class="emphasis"><em>x_max</em></span>.
 141         </p>
 142 <p>
 143           Requires <span class="emphasis"><em>x_min &lt; x_max</em></span>, otherwise <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 144           is called.
 145         </p>
 146 <p>
 147           For example:
 148         </p>
 149 <pre class="programlisting"><span class="identifier">arcsine_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">myarcsine</span><span class="special">(-</span><span class="number">2</span><span class="special">,</span> <span class="number">4</span><span class="special">);</span>
 150 </pre>
 151 <p>
 152           constructs an arcsine distribution with <span class="emphasis"><em>x_min = -2</em></span>
 153           and <span class="emphasis"><em>x_max = 4</em></span>.
 154         </p>
 155 <p>
 156           Default values of <span class="emphasis"><em>x_min = 0</em></span> and <span class="emphasis"><em>x_max =
 157           1</em></span> and a <code class="computeroutput"> <span class="keyword">typedef</span> <span class="identifier">arcsine_distribution</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">arcsine</span><span class="special">;</span></code>
 158           mean that
 159         </p>
 160 <pre class="programlisting"><span class="identifier">arcsine</span> <span class="identifier">as</span><span class="special">;</span>
 161 </pre>
 162 <p>
 163           constructs a 'Standard 01' arcsine distribution.
 164         </p>
 165 <h6>
 166 <a name="math_toolkit.dist_ref.dists.arcine_dist.h1"></a>
 167           <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.parameter_accessors"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.parameter_accessors">Parameter
 168           Accessors</a>
 169         </h6>
 170 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">x_min</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
 171 <span class="identifier">RealType</span> <span class="identifier">x_max</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
 172 </pre>
 173 <p>
 174           Return the parameter <span class="emphasis"><em>x_min</em></span> or <span class="emphasis"><em>x_max</em></span>
 175           from which this distribution was constructed.
 176         </p>
 177 <p>
 178           So, for example:
 179         </p>
 180 <pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">arcsine_distribution</span><span class="special">;</span>
 181
 182 <span class="identifier">arcsine_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">as</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">5</span><span class="special">);</span> <span class="comment">// Cconstructs a double arcsine distribution.</span>
 183 <span class="identifier">BOOST_ASSERT</span><span class="special">(</span><span class="identifier">as</span><span class="special">.</span><span class="identifier">x_min</span><span class="special">()</span> <span class="special">==</span> <span class="number">2.</span><span class="special">);</span>  <span class="comment">// as.x_min() returns 2.</span>
 184 <span class="identifier">BOOST_ASSERT</span><span class="special">(</span><span class="identifier">as</span><span class="special">.</span><span class="identifier">x_max</span><span class="special">()</span> <span class="special">==</span> <span class="number">5.</span><span class="special">);</span>   <span class="comment">// as.x_max()  returns 5.</span>
 185 </pre>
 186 <h5>
 187 <a name="math_toolkit.dist_ref.dists.arcine_dist.h2"></a>
 188           <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.non_member_accessor_functions"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.non_member_accessor_functions">Non-member
 189           Accessor Functions</a>
 190         </h5>
 191 <p>
 192           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 193           functions</a> that are generic to all distributions are supported:
 194           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 195           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 196           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
 197           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
 198           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
 199           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 200           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
 201           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
 202         </p>
 203 <p>
 204           The formulae for calculating these are shown in the table below, and at
 205           <a href="http://mathworld.wolfram.com/arcsineDistribution.html" target="_top">Wolfram
 206           Mathworld</a>.
 207         </p>
 208 <div class="note"><table border="0" summary="Note">
 209 <tr>
 210 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 211 <th align="left">Note</th>
 212 </tr>
 213 <tr><td align="left" valign="top"><p>
 214             There are always <span class="bold"><strong>two</strong></span> values for the
 215             <span class="bold"><strong>mode</strong></span>, at <span class="emphasis"><em>x_min</em></span>
 216             and at <span class="emphasis"><em>x_max</em></span>, default 0 and 1, so instead we raise
 217             the exception <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 218             At these extrema, the PDFs are infinite, and the CDFs zero or unity.
 219           </p></td></tr>
 220 </table></div>
 221 <h5>
 222 <a name="math_toolkit.dist_ref.dists.arcine_dist.h3"></a>
 223           <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.applications"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.applications">Applications</a>
 224         </h5>
 225 <p>
 226           The arcsine distribution is useful to describe <a href="http://en.wikipedia.org/wiki/Random_walk" target="_top">Random
 227           walks</a>, (including drunken walks) <a href="http://en.wikipedia.org/wiki/Brownian_motion" target="_top">Brownian
 228           motion</a>, <a href="http://en.wikipedia.org/wiki/Wiener_process" target="_top">Weiner
 229           processes</a>, <a href="http://en.wikipedia.org/wiki/Bernoulli_trial" target="_top">Bernoulli
 230           trials</a>, and their appplication to solve stock market and other
 231           <a href="http://en.wikipedia.org/wiki/Gambler%27s_ruin" target="_top">ruinous gambling
 232           games</a>.
 233         </p>
 234 <p>
 235           The random variate <span class="emphasis"><em>x</em></span> is constrained to <span class="emphasis"><em>x_min</em></span>
 236           and <span class="emphasis"><em>x_max</em></span>, (for our 'standard' distribution, 0 and
 237           1), and is usually some fraction. For any other <span class="emphasis"><em>x_min</em></span>
 238           and <span class="emphasis"><em>x_max</em></span> a fraction can be obtained from <span class="emphasis"><em>x</em></span>
 239           using
 240         </p>
 241 <div class="blockquote"><blockquote class="blockquote"><p>
 242             <span class="serif_italic">fraction = (x - x_min) / (x_max - x_min)</span>
 243           </p></blockquote></div>
 244 <p>
 245           The simplest example is tossing heads and tails with a fair coin and modelling
 246           the risk of losing, or winning. Walkers (molecules, drunks...) moving left
 247           or right of a centre line are another common example.
 248         </p>
 249 <p>
 250           The random variate <span class="emphasis"><em>x</em></span> is the fraction of time spent
 251           on the 'winning' side. If half the time is spent on the 'winning' side
 252           (and so the other half on the 'losing' side) then <span class="emphasis"><em>x = 1/2</em></span>.
 253         </p>
 254 <p>
 255           For large numbers of tosses, this is modelled by the (standard [0,1]) arcsine
 256           distribution, and the PDF can be calculated thus:
 257         </p>
 258 <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">1.</span> <span class="special">/</span> <span class="number">2</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.637</span>
 259 <span class="comment">// pdf has a minimum at x = 0.5</span>
 260 </pre>
 261 <p>
 262           From the plot of PDF, it is clear that <span class="emphasis"><em>x</em></span> = &#189; is the
 263           <span class="bold"><strong>minimum</strong></span> of the curve, so this is the
 264           <span class="bold"><strong>least likely</strong></span> scenario. (This is highly
 265           counter-intuitive, considering that fair tosses must <span class="bold"><strong>eventually</strong></span>
 266           become equal. It turns out that <span class="emphasis"><em>eventually</em></span> is not
 267           just very long, but <span class="bold"><strong>infinite</strong></span>!).
 268         </p>
 269 <p>
 270           The <span class="bold"><strong>most likely</strong></span> scenarios are towards
 271           the extrema where <span class="emphasis"><em>x</em></span> = 0 or <span class="emphasis"><em>x</em></span>
 272           = 1.
 273         </p>
 274 <p>
 275           If fraction of time on the left is a &#188;, it is only slightly more likely
 276           because the curve is quite flat bottomed.
 277         </p>
 278 <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">1.</span> <span class="special">/</span> <span class="number">4</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.735</span>
 279 </pre>
 280 <p>
 281           If we consider fair coin-tossing games being played for 100 days (hypothetically
 282           continuously to be 'at-limit') the person winning after day 5 will not
 283           change in fraction 0.144 of the cases.
 284         </p>
 285 <p>
 286           We can easily compute this setting <span class="emphasis"><em>x</em></span> = 5./100 = 0.05
 287         </p>
 288 <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">0.05</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.144</span>
 289 </pre>
 290 <p>
 291           Similarly, we can compute from a fraction of 0.05 /2 = 0.025 (halved because
 292           we are considering both winners and losers) corresponding to 1 - 0.025
 293           or 97.5% of the gamblers, (walkers, particles...) on the <span class="bold"><strong>same
 294           side</strong></span> of the origin
 295         </p>
 296 <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="number">2</span> <span class="special">*</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">1</span> <span class="special">-</span> <span class="number">0.975</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.202</span>
 297 </pre>
 298 <p>
 299           (use of the complement gives a bit more clarity, and avoids potential loss
 300           of accuracy when <span class="emphasis"><em>x</em></span> is close to unity, see <a class="link" href="../../stat_tut/overview/complements.html#why_complements">why
 301           complements?</a>).
 302         </p>
 303 <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="number">2</span> <span class="special">*</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">0.975</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.202</span>
 304 </pre>
 305 <p>
 306           or we can reverse the calculation by assuming a fraction of time on one
 307           side, say fraction 0.2,
 308         </p>
 309 <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">1</span> <span class="special">-</span> <span class="number">0.2</span> <span class="special">/</span> <span class="number">2</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">//  0.976</span>
 310
 311 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">0.2</span> <span class="special">/</span> <span class="number">2</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.976</span>
 312 </pre>
 313 <p>
 314           <span class="bold"><strong>Summary</strong></span>: Every time we toss, the odds
 315           are equal, so on average we have the same change of winning and losing.
 316         </p>
 317 <p>
 318           But this is <span class="bold"><strong>not true</strong></span> for an an individual
 319           game where one will be <span class="bold"><strong>mostly in a bad or good patch</strong></span>.
 320         </p>
 321 <p>
 322           This is quite counter-intuitive to most people, but the mathematics is
 323           clear, and gamblers continue to provide proof.
 324         </p>
 325 <p>
 326           <span class="bold"><strong>Moral</strong></span>: if you in a losing patch, leave
 327           the game. (Because the odds to recover to a good patch are poor).
 328         </p>
 329 <p>
 330           <span class="bold"><strong>Corollary</strong></span>: Quit while you are ahead?
 331         </p>
 332 <p>
 333           A working example is at <a href="../../../../../example/arcsine_example.cpp" target="_top">arcsine_example.cpp</a>
 334           including sample output .
 335         </p>
 336 <h5>
 337 <a name="math_toolkit.dist_ref.dists.arcine_dist.h4"></a>
 338           <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.related_distributions"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.related_distributions">Related
 339           distributions</a>
 340         </h5>
 341 <p>
 342           The arcsine distribution with <span class="emphasis"><em>x_min = 0</em></span> and <span class="emphasis"><em>x_max
 343           = 1</em></span> is special case of the <a class="link" href="beta_dist.html" title="Beta Distribution">Beta
 344           Distribution</a> with &#945; = 1/2 and &#946; = 1/2.
 345         </p>
 346 <h5>
 347 <a name="math_toolkit.dist_ref.dists.arcine_dist.h5"></a>
 348           <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.accuracy"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.accuracy">Accuracy</a>
 349         </h5>
 350 <p>
 351           This distribution is implemented using sqrt, sine, cos and arc sine and
 352           cos trigonometric functions which are normally accurate to a few <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">machine epsilon</a>.
 353           But all values suffer from <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">loss
 354           of significance or cancellation error</a> for values of <span class="emphasis"><em>x</em></span>
 355           close to <span class="emphasis"><em>x_max</em></span>. For example, for a standard [0, 1]
 356           arcsine distribution <span class="emphasis"><em>as</em></span>, the pdf is symmetric about
 357           random variate <span class="emphasis"><em>x = 0.5</em></span> so that one would expect <code class="computeroutput"><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">0.01</span><span class="special">)</span> <span class="special">==</span>
 358           <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">0.99</span><span class="special">)</span></code>. But
 359           as <span class="emphasis"><em>x</em></span> nears unity, there is increasing <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">loss
 360           of significance</a>. To counteract this, the complement versions of
 361           CDF and quantile are implemented with alternative expressions using <span class="emphasis"><em>cos<sup>-1</sup></em></span>
 362           instead of <span class="emphasis"><em>sin<sup>-1</sup></em></span>. Users should see <a class="link" href="../../stat_tut/overview/complements.html#why_complements">why
 363           complements?</a> for guidance on when to avoid loss of accuracy by using
 364           complements.
 365         </p>
 366 <h5>
 367 <a name="math_toolkit.dist_ref.dists.arcine_dist.h6"></a>
 368           <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.testing"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.testing">Testing</a>
 369         </h5>
 370 <p>
 371           The results were tested against a few accurate spot values computed by
 372           <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a>, for example:
 373         </p>
 374 <pre class="programlisting"><span class="identifier">N</span><span class="special">[</span><span class="identifier">PDF</span><span class="special">[</span><span class="identifier">arcsinedistribution</span><span class="special">[</span><span class="number">0</span><span class="special">,</span> <span class="number">1</span><span class="special">],</span> <span class="number">0.5</span><span class="special">],</span> <span class="number">50</span><span class="special">]</span>
 375 <span class="number">0.63661977236758134307553505349005744813783858296183</span>
 376 </pre>
 377 <h5>
 378 <a name="math_toolkit.dist_ref.dists.arcine_dist.h7"></a>
 379           <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.implementation"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.implementation">Implementation</a>
 380         </h5>
 381 <p>
 382           In the following table <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span>
 383           are the parameters <span class="emphasis"><em>x_min</em></span> and <span class="emphasis"><em>x_max</em></span>,
 384           <span class="emphasis"><em>x</em></span> is the random variable, <span class="emphasis"><em>p</em></span> is
 385           the probability and its complement <span class="emphasis"><em>q = 1-p</em></span>.
 386         </p>
 387 <div class="informaltable"><table class="table">
 388 <colgroup>
 389 <col>
 390 <col>
 391 </colgroup>
 392 <thead><tr>
 393 <th>
 394                   <p>
 395                     Function
 396                   </p>
 397                 </th>
 398 <th>
 399                   <p>
 400                     Implementation Notes
 401                   </p>
 402                 </th>
 403 </tr></thead>
 404 <tbody>
 405 <tr>
 406 <td>
 407                   <p>
 408                     support
 409                   </p>
 410                 </td>
 411 <td>
 412                   <p>
 413                     x &#8712; [a, b], default x &#8712; [0, 1]
 414                   </p>
 415                 </td>
 416 </tr>
 417 <tr>
 418 <td>
 419                   <p>
 420                     pdf
 421                   </p>
 422                 </td>
 423 <td>
 424                   <p>
 425                     f(x; a, b) = 1/(&#960;&#8901;&#8730;(x - a)&#8901;(b - x))
 426                   </p>
 427                 </td>
 428 </tr>
 429 <tr>
 430 <td>
 431                   <p>
 432                     cdf
 433                   </p>
 434                 </td>
 435 <td>
 436                   <p>
 437                     F(x) = 2/&#960;&#8901;sin<sup>-1</sup>(&#8730;(x - a) / (b - a) )
 438                   </p>
 439                 </td>
 440 </tr>
 441 <tr>
 442 <td>
 443                   <p>
 444                     cdf of complement
 445                   </p>
 446                 </td>
 447 <td>
 448                   <p>
 449                     2/(&#960;&#8901;cos<sup>-1</sup>(&#8730;(x - a) / (b - a)))
 450                   </p>
 451                 </td>
 452 </tr>
 453 <tr>
 454 <td>
 455                   <p>
 456                     quantile
 457                   </p>
 458                 </td>
 459 <td>
 460                   <p>
 461                     -a&#8901;sin<sup>2</sup>(&#189;&#960;&#8901;p) + a + b&#8901;sin<sup>2</sup>(&#189;&#960;&#8901;p)
 462                   </p>
 463                 </td>
 464 </tr>
 465 <tr>
 466 <td>
 467                   <p>
 468                     quantile from the complement
 469                   </p>
 470                 </td>
 471 <td>
 472                   <p>
 473                     -a&#8901;cos<sup>2</sup>(&#189;&#960;&#8901;p) + a + b&#8901;cos<sup>2</sup>(&#189;&#960;&#8901;q)
 474                   </p>
 475                 </td>
 476 </tr>
 477 <tr>
 478 <td>
 479                   <p>
 480                     mean
 481                   </p>
 482                 </td>
 483 <td>
 484                   <p>
 485                     &#189;(a+b)
 486                   </p>
 487                 </td>
 488 </tr>
 489 <tr>
 490 <td>
 491                   <p>
 492                     median
 493                   </p>
 494                 </td>
 495 <td>
 496                   <p>
 497                     &#189;(a+b)
 498                   </p>
 499                 </td>
 500 </tr>
 501 <tr>
 502 <td>
 503                   <p>
 504                     mode
 505                   </p>
 506                 </td>
 507 <td>
 508                   <p>
 509                     x &#8712; [a, b], so raises domain_error (returning NaN).
 510                   </p>
 511                 </td>
 512 </tr>
 513 <tr>
 514 <td>
 515                   <p>
 516                     variance
 517                   </p>
 518                 </td>
 519 <td>
 520                   <p>
 521                     (b - a)<sup>2</sup> / 8
 522                   </p>
 523                 </td>
 524 </tr>
 525 <tr>
 526 <td>
 527                   <p>
 528                     skewness
 529                   </p>
 530                 </td>
 531 <td>
 532                   <p>
 533                     0
 534                   </p>
 535                 </td>
 536 </tr>
 537 <tr>
 538 <td>
 539                   <p>
 540                     kurtosis excess
 541                   </p>
 542                 </td>
 543 <td>
 544                   <p>
 545                     -3/2
 546                   </p>
 547                 </td>
 548 </tr>
 549 <tr>
 550 <td>
 551                   <p>
 552                     kurtosis
 553                   </p>
 554                 </td>
 555 <td>
 556                   <p>
 557                     kurtosis_excess + 3
 558                   </p>
 559                 </td>
 560 </tr>
 561 </tbody>
 562 </table></div>
 563 <p>
 564           The quantile was calculated using an expression obtained by using <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a> to invert the
 565           formula for the CDF thus
 566         </p>
 567 <pre class="programlisting"><span class="identifier">solve</span> <span class="special">[</span><span class="identifier">p</span> <span class="special">-</span> <span class="number">2</span><span class="special">/</span><span class="identifier">pi</span> <span class="identifier">sin</span><span class="special">^-</span><span class="number">1</span><span class="special">(</span><span class="identifier">sqrt</span><span class="special">((</span><span class="identifier">x</span><span class="special">-</span><span class="identifier">a</span><span class="special">)/(</span><span class="identifier">b</span><span class="special">-</span><span class="identifier">a</span><span class="special">)))</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">x</span><span class="special">]</span>
 568 </pre>
 569 <p>
 570           which was interpreted as
 571         </p>
 572 <pre class="programlisting"><span class="identifier">Solve</span><span class="special">[</span><span class="identifier">p</span> <span class="special">-</span> <span class="special">(</span><span class="number">2</span> <span class="identifier">ArcSin</span><span class="special">[</span><span class="identifier">Sqrt</span><span class="special">[(-</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">x</span><span class="special">)/(-</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span><span class="special">)]])/</span><span class="identifier">Pi</span> <span class="special">==</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">MaxExtraConditions</span> <span class="special">-&gt;</span> <span class="identifier">Automatic</span><span class="special">]</span>
 573 </pre>
 574 <p>
 575           and produced the resulting expression
 576         </p>
 577 <div class="blockquote"><blockquote class="blockquote"><p>
 578             <span class="serif_italic">x = -a sin^2((pi p)/2)+a+b sin^2((pi p)/2)</span>
 579           </p></blockquote></div>
 580 <p>
 581           Thanks to Wolfram for providing this facility.
 582         </p>
 583 <h5>
 584 <a name="math_toolkit.dist_ref.dists.arcine_dist.h8"></a>
 585           <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.references"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.references">References</a>
 586         </h5>
 587 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 588 <li class="listitem">
 589               <a href="http://en.wikipedia.org/wiki/arcsine_distribution" target="_top">Wikipedia
 590               arcsine distribution</a>
 591             </li>
 592 <li class="listitem">
 593               <a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">Wikipedia
 594               Beta distribution</a>
 595             </li>
 596 <li class="listitem">
 597               <a href="http://mathworld.wolfram.com/BetaDistribution.html" target="_top">Wolfram
 598               MathWorld</a>
 599             </li>
 600 <li class="listitem">
 601               <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a>
 602             </li>
 603 </ul></div>
 604 <h5>
 605 <a name="math_toolkit.dist_ref.dists.arcine_dist.h9"></a>
 606           <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.sources"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.sources">Sources</a>
 607         </h5>
 608 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 609 <li class="listitem">
 610               <a href="http://estebanmoro.org/2009/04/the-probability-of-going-through-a-bad-patch" target="_top">The
 611               probability of going through a bad patch</a> Esteban Moro's Blog.
 612             </li>
 613 <li class="listitem">
 614               <a href="http://www.gotohaggstrom.com/What%20do%20schmucks%20and%20the%20arc%20sine%20law%20have%20in%20common.pdf" target="_top">What
 615               soschumcks and the arc sine have in common</a> Peter Haggstrom.
 616             </li>
 617 <li class="listitem">
 618               <a href="http://www.math.uah.edu/stat/special/Arcsine.html" target="_top">arcsine
 619               distribution</a>.
 620             </li>
 621 <li class="listitem">
 622               <a href="http://reference.wolfram.com/language/ref/ArcSinDistribution.html" target="_top">Wolfram
 623               reference arcsine examples</a>.
 624             </li>
 625 <li class="listitem">
 626               <a href="http://www.math.harvard.edu/library/sternberg/slides/1180908.pdf" target="_top">Shlomo
 627               Sternberg slides</a>.
 628             </li>
 629 </ul></div>
 630 </div>
 631 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 632 <td align="left"></td>
 633 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 634       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 635       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 636       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 637       Daryle Walker and Xiaogang Zhang<p>
 638         Distributed under the Boost Software License, Version 1.0. (See accompanying
 639         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
 640       </p>
 641 </div></td>
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