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  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.stat_tut.overview.generic"></a><a class="link" href="generic.html" title="Generic operations common to all distributions are non-member functions">Generic operations
  28         common to all distributions are non-member functions</a>
  29 </h4></div></div></div>
  30 <p>
  31           Want to calculate the PDF (Probability Density Function) of a distribution?
  32           No problem, just use:
  33         </p>
  34 <pre class="programlisting"><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">my_dist</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span>  <span class="comment">// Returns PDF (density) at point x of distribution my_dist.</span>
  35 </pre>
  36 <p>
  37           Or how about the CDF (Cumulative Distribution Function):
  38         </p>
  39 <pre class="programlisting"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">my_dist</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span>  <span class="comment">// Returns CDF (integral from -infinity to point x)</span>
  40                   <span class="comment">// of distribution my_dist.</span>
  41 </pre>
  42 <p>
  43           And quantiles are just the same:
  44         </p>
  45 <pre class="programlisting"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">my_dist</span><span class="special">,</span> <span class="identifier">p</span><span class="special">);</span>  <span class="comment">// Returns the value of the random variable x</span>
  46                        <span class="comment">// such that cdf(my_dist, x) == p.</span>
  47 </pre>
  48 <p>
  49           If you're wondering why these aren't member functions, it's to make the
  50           library more easily extensible: if you want to add additional generic operations
  51           - let's say the <span class="emphasis"><em>n'th moment</em></span> - then all you have to
  52           do is add the appropriate non-member functions, overloaded for each implemented
  53           distribution type.
  54         </p>
  55 <div class="tip"><table border="0" summary="Tip">
  56 <tr>
  57 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../../doc/src/images/tip.png"></td>
  58 <th align="left">Tip</th>
  59 </tr>
  60 <tr><td align="left" valign="top">
  61 <p>
  62             <span class="bold"><strong>Random numbers that approximate Quantiles of Distributions</strong></span>
  63           </p>
  64 <p>
  65             If you want random numbers that are distributed in a specific way, for
  66             example in a uniform, normal or triangular, see <a href="http://www.boost.org/libs/random/" target="_top">Boost.Random</a>.
  67           </p>
  68 <p>
  69             Whilst in principal there's nothing to prevent you from using the quantile
  70             function to convert a uniformly distributed random number to another
  71             distribution, in practice there are much more efficient algorithms available
  72             that are specific to random number generation.
  73           </p>
  74 </td></tr>
  75 </table></div>
  76 <p>
  77           For example, the binomial distribution has two parameters: n (the number
  78           of trials) and p (the probability of success on any one trial).
  79         </p>
  80 <p>
  81           The <code class="computeroutput"><span class="identifier">binomial_distribution</span></code>
  82           constructor therefore has two parameters:
  83         </p>
  84 <p>
  85           <code class="computeroutput"><span class="identifier">binomial_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">RealType</span>
  86           <span class="identifier">p</span><span class="special">);</span></code>
  87         </p>
  88 <p>
  89           For this distribution the <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random
  90           variate</a> is k: the number of successes observed. The probability
  91           density/mass function (pdf) is therefore written as <span class="emphasis"><em>f(k; n, p)</em></span>.
  92         </p>
  93 <div class="note"><table border="0" summary="Note">
  94 <tr>
  95 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
  96 <th align="left">Note</th>
  97 </tr>
  98 <tr><td align="left" valign="top">
  99 <p>
 100             <span class="bold"><strong>Random Variates and Distribution Parameters</strong></span>
 101           </p>
 102 <p>
 103             The concept of a <a href="http://en.wikipedia.org/wiki/Random_variable" target="_top">random
 104             variable</a> is closely linked to the term <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random
 105             variate</a>: a random variate is a particular value (outcome) of
 106             a random variable. and <a href="http://en.wikipedia.org/wiki/Parameter" target="_top">distribution
 107             parameters</a> are conventionally distinguished (for example in Wikipedia
 108             and Wolfram MathWorld) by placing a semi-colon or vertical bar) <span class="emphasis"><em>after</em></span>
 109             the <a href="http://en.wikipedia.org/wiki/Random_variable" target="_top">random
 110             variable</a> (whose value you 'choose'), to separate the variate
 111             from the parameter(s) that defines the shape of the distribution.
 112           </p>
 113 <p>
 114             For example, the binomial distribution probability distribution function
 115             (PDF) is written as <span class="serif_italic"><span class="emphasis"><em>f(k| n, p)</em></span>
 116             = Pr(K = k|n, p) = </span> probability of observing k successes out
 117             of n trials. K is the <a href="http://en.wikipedia.org/wiki/Random_variable" target="_top">random
 118             variable</a>, k is the <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random
 119             variate</a>, the parameters are n (trials) and p (probability).
 120           </p>
 121 </td></tr>
 122 </table></div>
 123 <div class="note"><table border="0" summary="Note">
 124 <tr>
 125 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 126 <th align="left">Note</th>
 127 </tr>
 128 <tr><td align="left" valign="top"><p>
 129             By convention, <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random
 130             variate</a> are lower case, usually k is integral, x if real, and
 131             <a href="http://en.wikipedia.org/wiki/Random_variable" target="_top">random variable</a>
 132             are upper case, K if integral, X if real. But this implementation treats
 133             all as floating point values <code class="computeroutput"><span class="identifier">RealType</span></code>,
 134             so if you really want an integral result, you must round: see note on
 135             Discrete Probability Distributions below for details.
 136           </p></td></tr>
 137 </table></div>
 138 <p>
 139           As noted above the non-member function <code class="computeroutput"><span class="identifier">pdf</span></code>
 140           has one parameter for the distribution object, and a second for the random
 141           variate. So taking our binomial distribution example, we would write:
 142         </p>
 143 <p>
 144           <code class="computeroutput"><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">binomial_distribution</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;(</span><span class="identifier">n</span><span class="special">,</span> <span class="identifier">p</span><span class="special">),</span> <span class="identifier">k</span><span class="special">);</span></code>
 145         </p>
 146 <p>
 147           The ranges of <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random
 148           variate</a> values that are permitted and are supported can be tested
 149           by using two functions <code class="computeroutput"><span class="identifier">range</span></code>
 150           and <code class="computeroutput"><span class="identifier">support</span></code>.
 151         </p>
 152 <p>
 153           The distribution (effectively the <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random
 154           variate</a>) is said to be 'supported' over a range that is <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">"the smallest
 155           closed set whose complement has probability zero"</a>. MathWorld
 156           uses the word 'defined' for this range. Non-mathematicians might say it
 157           means the 'interesting' smallest range of random variate x that has the
 158           cdf going from zero to unity. Outside are uninteresting zones where the
 159           pdf is zero, and the cdf zero or unity.
 160         </p>
 161 <p>
 162           For most distributions, with probability distribution functions one might
 163           describe as 'well-behaved', we have decided that it is most useful for
 164           the supported range to <span class="bold"><strong>exclude</strong></span> random
 165           variate values like exact zero <span class="bold"><strong>if the end point is
 166           discontinuous</strong></span>. For example, the Weibull (scale 1, shape 1) distribution
 167           smoothly heads for unity as the random variate x declines towards zero.
 168           But at x = zero, the value of the pdf is suddenly exactly zero, by definition.
 169           If you are plotting the PDF, or otherwise calculating, zero is not the
 170           most useful value for the lower limit of supported, as we discovered. So
 171           for this, and similar distributions, we have decided it is most numerically
 172           useful to use the closest value to zero, min_value, for the limit of the
 173           supported range. (The <code class="computeroutput"><span class="identifier">range</span></code>
 174           remains from zero, so you will still get <code class="computeroutput"><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">weibull</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span>
 175           <span class="special">==</span> <span class="number">0</span></code>).
 176           (Exponential and gamma distributions have similarly discontinuous functions).
 177         </p>
 178 <p>
 179           Mathematically, the functions may make sense with an (+ or -) infinite
 180           value, but except for a few special cases (in the Normal and Cauchy distributions)
 181           this implementation limits random variates to finite values from the <code class="computeroutput"><span class="identifier">max</span></code> to <code class="computeroutput"><span class="identifier">min</span></code>
 182           for the <code class="computeroutput"><span class="identifier">RealType</span></code>. (See
 183           <a class="link" href="../../sf_implementation.html#math_toolkit.sf_implementation.handling_of_floating_point_infin">Handling
 184           of Floating-Point Infinity</a> for rationale).
 185         </p>
 186 <div class="note"><table border="0" summary="Note">
 187 <tr>
 188 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 189 <th align="left">Note</th>
 190 </tr>
 191 <tr><td align="left" valign="top">
 192 <p>
 193             <span class="bold"><strong>Discrete Probability Distributions</strong></span>
 194           </p>
 195 <p>
 196             Note that the <a href="http://en.wikipedia.org/wiki/Discrete_probability_distribution" target="_top">discrete
 197             distributions</a>, including the binomial, negative binomial, Poisson
 198             &amp; Bernoulli, are all mathematically defined as discrete functions:
 199             that is to say the functions <code class="computeroutput"><span class="identifier">cdf</span></code>
 200             and <code class="computeroutput"><span class="identifier">pdf</span></code> are only defined
 201             for integral values of the random variate.
 202           </p>
 203 <p>
 204             However, because the method of calculation often uses continuous functions
 205             it is convenient to treat them as if they were continuous functions,
 206             and permit non-integral values of their parameters.
 207           </p>
 208 <p>
 209             Users wanting to enforce a strict mathematical model may use <code class="computeroutput"><span class="identifier">floor</span></code> or <code class="computeroutput"><span class="identifier">ceil</span></code>
 210             functions on the random variate prior to calling the distribution function.
 211           </p>
 212 <p>
 213             The quantile functions for these distributions are hard to specify in
 214             a manner that will satisfy everyone all of the time. The default behaviour
 215             is to return an integer result, that has been rounded <span class="emphasis"><em>outwards</em></span>:
 216             that is to say, lower quantiles - where the probablity is less than 0.5
 217             are rounded down, while upper quantiles - where the probability is greater
 218             than 0.5 - are rounded up. This behaviour ensures that if an X% quantile
 219             is requested, then <span class="emphasis"><em>at least</em></span> the requested coverage
 220             will be present in the central region, and <span class="emphasis"><em>no more than</em></span>
 221             the requested coverage will be present in the tails.
 222           </p>
 223 <p>
 224             This behaviour can be changed so that the quantile functions are rounded
 225             differently, or return a real-valued result using <a class="link" href="../../pol_overview.html" title="Policy Overview">Policies</a>.
 226             It is strongly recommended that you read the tutorial <a class="link" href="../../pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">Understanding
 227             Quantiles of Discrete Distributions</a> before using the quantile
 228             function on a discrete distribtion. The <a class="link" href="../../pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">reference
 229             docs</a> describe how to change the rounding policy for these distributions.
 230           </p>
 231 <p>
 232             For similar reasons continuous distributions with parameters like "degrees
 233             of freedom" that might appear to be integral, are treated as real
 234             values (and are promoted from integer to floating-point if necessary).
 235             In this case however, there are a small number of situations where non-integral
 236             degrees of freedom do have a genuine meaning.
 237           </p>
 238 </td></tr>
 239 </table></div>
 240 </div>
 241 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 242 <td align="left"></td>
 243 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 244       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 245       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 246       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 247       Daryle Walker and Xiaogang Zhang<p>
 248         Distributed under the Boost Software License, Version 1.0. (See accompanying
 249         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>)
 250       </p>
 251 </div></td>
 252 </tr></table>
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