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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.pol_tutorial.understand_dis_quant"></a><a class="link" href="understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">Understanding
  28       Quantiles of Discrete Distributions</a>
  29 </h3></div></div></div>
  30 <p>
  31         Discrete distributions present us with a problem when calculating the quantile:
  32         we are starting from a continuous real-valued variable - the probability
  33         - but the result (the value of the random variable) should really be discrete.
  34       </p>
  35 <p>
  36         Consider for example a Binomial distribution, with a sample size of 50, and
  37         a success fraction of 0.5. There are a variety of ways we can plot a discrete
  38         distribution, but if we plot the PDF as a step-function then it looks something
  39         like this:
  40       </p>
  41 <p>
  42         <span class="inlinemediaobject"><img src="../../../graphs/binomial_pdf.png"></span>
  43       </p>
  44 <p>
  45         Now lets suppose that the user asks for a the quantile that corresponds to
  46         a probability of 0.05, if we zoom in on the CDF for that region here's what
  47         we see:
  48       </p>
  49 <p>
  50         <span class="inlinemediaobject"><img src="../../../graphs/binomial_quantile_1.png"></span>
  51       </p>
  52 <p>
  53         As can be seen there is no random variable that corresponds to a probability
  54         of exactly 0.05, so we're left with two choices as shown in the figure:
  55       </p>
  56 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  57 <li class="listitem">
  58             We could round the result down to 18.
  59           </li>
  60 <li class="listitem">
  61             We could round the result up to 19.
  62           </li>
  63 </ul></div>
  64 <p>
  65         In fact there's actually a third choice as well: we could "pretend"
  66         that the distribution was continuous and return a real valued result: in
  67         this case we would calculate a result of approximately 18.701 (this accurately
  68         reflects the fact that the result is nearer to 19 than 18).
  69       </p>
  70 <p>
  71         By using policies we can offer any of the above as options, but that still
  72         leaves the question: <span class="emphasis"><em>What is actually the right thing to do?</em></span>
  73       </p>
  74 <p>
  75         And in particular: <span class="emphasis"><em>What policy should we use by default?</em></span>
  76       </p>
  77 <p>
  78         In coming to an answer we should realise that:
  79       </p>
  80 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  81 <li class="listitem">
  82             Calculating an integer result is often much faster than calculating a
  83             real-valued result: in fact in our tests it was up to 20 times faster.
  84           </li>
  85 <li class="listitem">
  86             Normally people calculate quantiles so that they can perform a test of
  87             some kind: <span class="emphasis"><em>"If the random variable is less than N then
  88             we can reject our null-hypothesis with 90% confidence."</em></span>
  89           </li>
  90 </ul></div>
  91 <p>
  92         So there is a genuine benefit to calculating an integer result as well as
  93         it being "the right thing to do" from a philosophical point of
  94         view. What's more if someone asks for a quantile at 0.05, then we can normally
  95         assume that they are asking for <span class="emphasis"><em><span class="bold"><strong>at least</strong></span>
  96         95% of the probability to the right of the value chosen, and <span class="bold"><strong>no
  97         more than</strong></span> 5% of the probability to the left of the value chosen.</em></span>
  98       </p>
  99 <p>
 100         In the above binomial example we would therefore round the result down to
 101         18.
 102       </p>
 103 <p>
 104         The converse applies to upper-quantiles: If the probability is greater than
 105         0.5 we would want to round the quantile up, <span class="emphasis"><em>so that <span class="bold"><strong>at
 106         least</strong></span> the requested probability is to the left of the value returned,
 107         and <span class="bold"><strong>no more than</strong></span> 1 - the requested probability
 108         is to the right of the value returned.</em></span>
 109       </p>
 110 <p>
 111         Likewise for two-sided intervals, we would round lower quantiles down, and
 112         upper quantiles up. This ensures that we have <span class="emphasis"><em>at least the requested
 113         probability in the central region</em></span> and <span class="emphasis"><em>no more than 1
 114         minus the requested probability in the tail areas.</em></span>
 115       </p>
 116 <p>
 117         For example, taking our 50 sample binomial distribution with a success fraction
 118         of 0.5, if we wanted a two sided 90% confidence interval, then we would ask
 119         for the 0.05 and 0.95 quantiles with the results <span class="emphasis"><em>rounded outwards</em></span>
 120         so that <span class="emphasis"><em>at least 90% of the probability</em></span> is in the central
 121         area:
 122       </p>
 123 <p>
 124         <span class="inlinemediaobject"><img src="../../../graphs/binomial_pdf_3.png"></span>
 125       </p>
 126 <p>
 127         So far so good, but there is in fact a trap waiting for the unwary here:
 128       </p>
 129 <pre class="programlisting"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binomial</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.05</span><span class="special">);</span>
 130 </pre>
 131 <p>
 132         returns 18 as the result, which is what we would expect from the graph above,
 133         and indeed there is no x greater than 18 for which:
 134       </p>
 135 <pre class="programlisting"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">binomial</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">&lt;=</span> <span class="number">0.05</span><span class="special">;</span>
 136 </pre>
 137 <p>
 138         However:
 139       </p>
 140 <pre class="programlisting"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binomial</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.95</span><span class="special">);</span>
 141 </pre>
 142 <p>
 143         returns 31, and indeed while there is no x less than 31 for which:
 144       </p>
 145 <pre class="programlisting"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">binomial</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">&gt;=</span> <span class="number">0.95</span><span class="special">;</span>
 146 </pre>
 147 <p>
 148         We might naively expect that for this symmetrical distribution the result
 149         would be 32 (since 32 = 50 - 18), but we need to remember that the cdf of
 150         the binomial is <span class="emphasis"><em>inclusive</em></span> of the random variable. So
 151         while the left tail area <span class="emphasis"><em>includes</em></span> the quantile returned,
 152         the right tail area always excludes an upper quantile value: since that "belongs"
 153         to the central area.
 154       </p>
 155 <p>
 156         Look at the graph above to see what's going on here: the lower quantile of
 157         18 belongs to the left tail, so any value &lt;= 18 is in the left tail. The
 158         upper quantile of 31 on the other hand belongs to the central area, so the
 159         tail area actually starts at 32, so any value &gt; 31 is in the right tail.
 160       </p>
 161 <p>
 162         Therefore if U and L are the upper and lower quantiles respectively, then
 163         a random variable X is in the tail area - where we would reject the null
 164         hypothesis if:
 165       </p>
 166 <pre class="programlisting"><span class="identifier">X</span> <span class="special">&lt;=</span> <span class="identifier">L</span> <span class="special">||</span> <span class="identifier">X</span> <span class="special">&gt;</span> <span class="identifier">U</span>
 167 </pre>
 168 <p>
 169         And the a variable X is inside the central region if:
 170       </p>
 171 <pre class="programlisting"><span class="identifier">L</span> <span class="special">&lt;</span> <span class="identifier">X</span> <span class="special">&lt;=</span> <span class="identifier">U</span>
 172 </pre>
 173 <p>
 174         The moral here is to <span class="emphasis"><em>always be very careful with your comparisons
 175         when dealing with a discrete distribution</em></span>, and if in doubt, <span class="emphasis"><em>base
 176         your comparisons on CDF's instead</em></span>.
 177       </p>
 178 <h5>
 179 <a name="math_toolkit.pol_tutorial.understand_dis_quant.h0"></a>
 180         <span class="phrase"><a name="math_toolkit.pol_tutorial.understand_dis_quant.other_rounding_policies_are_avai"></a></span><a class="link" href="understand_dis_quant.html#math_toolkit.pol_tutorial.understand_dis_quant.other_rounding_policies_are_avai">Other
 181         Rounding Policies are Available</a>
 182       </h5>
 183 <p>
 184         As you would expect from a section on policies, you won't be surprised to
 185         know that other rounding options are available:
 186       </p>
 187 <div class="variablelist">
 188 <p class="title"><b></b></p>
 189 <dl class="variablelist">
 190 <dt><span class="term">integer_round_outwards</span></dt>
 191 <dd>
 192 <p>
 193               This is the default policy as described above: lower quantiles are
 194               rounded down (probability &lt; 0.5), and upper quantiles (probability
 195               &gt; 0.5) are rounded up.
 196             </p>
 197 <p>
 198               This gives <span class="emphasis"><em>no more than</em></span> the requested probability
 199               in the tails, and <span class="emphasis"><em>at least</em></span> the requested probability
 200               in the central area.
 201             </p>
 202 </dd>
 203 <dt><span class="term">integer_round_inwards</span></dt>
 204 <dd>
 205 <p>
 206               This is the exact opposite of the default policy: lower quantiles are
 207               rounded up (probability &lt; 0.5), and upper quantiles (probability
 208               &gt; 0.5) are rounded down.
 209             </p>
 210 <p>
 211               This gives <span class="emphasis"><em>at least</em></span> the requested probability
 212               in the tails, and <span class="emphasis"><em>no more than</em></span> the requested probability
 213               in the central area.
 214             </p>
 215 </dd>
 216 <dt><span class="term">integer_round_down</span></dt>
 217 <dd><p>
 218               This policy will always round the result down no matter whether it
 219               is an upper or lower quantile
 220             </p></dd>
 221 <dt><span class="term">integer_round_up</span></dt>
 222 <dd><p>
 223               This policy will always round the result up no matter whether it is
 224               an upper or lower quantile
 225             </p></dd>
 226 <dt><span class="term">integer_round_nearest</span></dt>
 227 <dd><p>
 228               This policy will always round the result to the nearest integer no
 229               matter whether it is an upper or lower quantile
 230             </p></dd>
 231 <dt><span class="term">real</span></dt>
 232 <dd><p>
 233               This policy will return a real valued result for the quantile of a
 234               discrete distribution: this is generally much slower than finding an
 235               integer result but does allow for more sophisticated rounding policies.
 236             </p></dd>
 237 </dl>
 238 </div>
 239 <p>
 240         To understand how the rounding policies for the discrete distributions can
 241         be used, we'll use the 50-sample binomial distribution with a success fraction
 242         of 0.5 once again, and calculate all the possible quantiles at 0.05 and 0.95.
 243       </p>
 244 <p>
 245         Begin by including the needed headers (and some using statements for conciseness):
 246       </p>
 247 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
 248 <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 249 <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">left</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">fixed</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">scientific</span><span class="special">;</span>
 250 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
 251 <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">;</span>
 252 <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">;</span>
 253
 254 <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">binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
 255 </pre>
 256 <p>
 257         Next we'll bring the needed declarations into scope, and define distribution
 258         types for all the available rounding policies:
 259       </p>
 260 <pre class="programlisting"><span class="comment">// Avoid </span>
 261 <span class="comment">// using namespace std; // and </span>
 262 <span class="comment">// using namespace boost::math;</span>
 263 <span class="comment">// to avoid potential ambiguity of names, like binomial.</span>
 264 <span class="comment">// using namespace boost::math::policies; is small risk, but</span>
 265 <span class="comment">// the necessary items are brought into scope thus:</span>
 266
 267 <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">binomial_distribution</span><span class="special">;</span>
 268 <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">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special">;</span>
 269 <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">policies</span><span class="special">::</span><span class="identifier">discrete_quantile</span><span class="special">;</span>
 270
 271 <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">policies</span><span class="special">::</span><span class="identifier">integer_round_outwards</span><span class="special">;</span>
 272 <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">policies</span><span class="special">::</span><span class="identifier">integer_round_down</span><span class="special">;</span>
 273 <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">policies</span><span class="special">::</span><span class="identifier">integer_round_up</span><span class="special">;</span>
 274 <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">policies</span><span class="special">::</span><span class="identifier">integer_round_nearest</span><span class="special">;</span>
 275 <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">policies</span><span class="special">::</span><span class="identifier">integer_round_inwards</span><span class="special">;</span>
 276 <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">policies</span><span class="special">::</span><span class="identifier">real</span><span class="special">;</span>
 277
 278 <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">binomial_distribution</span><span class="special">;</span> <span class="comment">// Not std::binomial_distribution.</span>
 279
 280 <span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special">&lt;</span>
 281             <span class="keyword">double</span><span class="special">,</span>
 282             <span class="identifier">policy</span><span class="special">&lt;</span><span class="identifier">discrete_quantile</span><span class="special">&lt;</span><span class="identifier">integer_round_outwards</span><span class="special">&gt;</span> <span class="special">&gt;</span> <span class="special">&gt;</span>
 283         <span class="identifier">binom_round_outwards</span><span class="special">;</span>
 284
 285 <span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special">&lt;</span>
 286             <span class="keyword">double</span><span class="special">,</span>
 287             <span class="identifier">policy</span><span class="special">&lt;</span><span class="identifier">discrete_quantile</span><span class="special">&lt;</span><span class="identifier">integer_round_inwards</span><span class="special">&gt;</span> <span class="special">&gt;</span> <span class="special">&gt;</span>
 288         <span class="identifier">binom_round_inwards</span><span class="special">;</span>
 289
 290 <span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special">&lt;</span>
 291             <span class="keyword">double</span><span class="special">,</span>
 292             <span class="identifier">policy</span><span class="special">&lt;</span><span class="identifier">discrete_quantile</span><span class="special">&lt;</span><span class="identifier">integer_round_down</span><span class="special">&gt;</span> <span class="special">&gt;</span> <span class="special">&gt;</span>
 293         <span class="identifier">binom_round_down</span><span class="special">;</span>
 294
 295 <span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special">&lt;</span>
 296             <span class="keyword">double</span><span class="special">,</span>
 297             <span class="identifier">policy</span><span class="special">&lt;</span><span class="identifier">discrete_quantile</span><span class="special">&lt;</span><span class="identifier">integer_round_up</span><span class="special">&gt;</span> <span class="special">&gt;</span> <span class="special">&gt;</span>
 298         <span class="identifier">binom_round_up</span><span class="special">;</span>
 299
 300 <span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special">&lt;</span>
 301             <span class="keyword">double</span><span class="special">,</span>
 302             <span class="identifier">policy</span><span class="special">&lt;</span><span class="identifier">discrete_quantile</span><span class="special">&lt;</span><span class="identifier">integer_round_nearest</span><span class="special">&gt;</span> <span class="special">&gt;</span> <span class="special">&gt;</span>
 303         <span class="identifier">binom_round_nearest</span><span class="special">;</span>
 304
 305 <span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special">&lt;</span>
 306             <span class="keyword">double</span><span class="special">,</span>
 307             <span class="identifier">policy</span><span class="special">&lt;</span><span class="identifier">discrete_quantile</span><span class="special">&lt;</span><span class="identifier">real</span><span class="special">&gt;</span> <span class="special">&gt;</span> <span class="special">&gt;</span>
 308         <span class="identifier">binom_real_quantile</span><span class="special">;</span>
 309 </pre>
 310 <p>
 311         Now let's set to work calling those quantiles:
 312       </p>
 313 <pre class="programlisting"><span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
 314 <span class="special">{</span>
 315    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span>
 316       <span class="string">"Testing rounding policies for a 50 sample binomial distribution,\n"</span>
 317       <span class="string">"with a success fraction of 0.5.\n\n"</span>
 318       <span class="string">"Lower quantiles are calculated at p = 0.05\n\n"</span>
 319       <span class="string">"Upper quantiles at p = 0.95.\n\n"</span><span class="special">;</span>
 320
 321    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
 322       <span class="special">&lt;&lt;</span> <span class="string">"Policy"</span><span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
 323       <span class="special">&lt;&lt;</span> <span class="string">"Lower Quantile"</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
 324       <span class="special">&lt;&lt;</span> <span class="string">"Upper Quantile"</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 325
 326    <span class="comment">// Test integer_round_outwards:</span>
 327    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
 328       <span class="special">&lt;&lt;</span> <span class="string">"integer_round_outwards"</span>
 329       <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
 330       <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_round_outwards</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.05</span><span class="special">)</span>
 331       <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
 332       <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_round_outwards</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.95</span><span class="special">)</span>
 333       <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 334
 335    <span class="comment">// Test integer_round_inwards:</span>
 336    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
 337       <span class="special">&lt;&lt;</span> <span class="string">"integer_round_inwards"</span>
 338       <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
 339       <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_round_inwards</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.05</span><span class="special">)</span>
 340       <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
 341       <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_round_inwards</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.95</span><span class="special">)</span>
 342       <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 343
 344    <span class="comment">// Test integer_round_down:</span>
 345    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
 346       <span class="special">&lt;&lt;</span> <span class="string">"integer_round_down"</span>
 347       <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
 348       <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_round_down</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.05</span><span class="special">)</span>
 349       <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
 350       <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_round_down</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.95</span><span class="special">)</span>
 351       <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 352
 353    <span class="comment">// Test integer_round_up:</span>
 354    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
 355       <span class="special">&lt;&lt;</span> <span class="string">"integer_round_up"</span>
 356       <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
 357       <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_round_up</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.05</span><span class="special">)</span>
 358       <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
 359       <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_round_up</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.95</span><span class="special">)</span>
 360       <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 361
 362    <span class="comment">// Test integer_round_nearest:</span>
 363    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
 364       <span class="special">&lt;&lt;</span> <span class="string">"integer_round_nearest"</span>
 365       <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
 366       <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_round_nearest</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.05</span><span class="special">)</span>
 367       <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
 368       <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_round_nearest</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.95</span><span class="special">)</span>
 369       <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 370
 371    <span class="comment">// Test real:</span>
 372    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
 373       <span class="special">&lt;&lt;</span> <span class="string">"real"</span>
 374       <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
 375       <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_real_quantile</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.05</span><span class="special">)</span>
 376       <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
 377       <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_real_quantile</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.95</span><span class="special">)</span>
 378       <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 379 <span class="special">}</span> <span class="comment">// int main()</span>
 380 </pre>
 381 <p>
 382         Which produces the program output:
 383       </p>
 384 <pre class="programlisting">  policy_eg_10.vcxproj -&gt; J:\Cpp\MathToolkit\test\Math_test\Release\policy_eg_10.exe
 385   Testing rounding policies for a 50 sample binomial distribution,
 386   with a success fraction of 0.5.
 387
 388   Lower quantiles are calculated at p = 0.05
 389
 390   Upper quantiles at p = 0.95.
 391
 392                      Policy    Lower Quantile    Upper Quantile
 393      integer_round_outwards                18                31
 394       integer_round_inwards                19                30
 395          integer_round_down                18                30
 396            integer_round_up                19                31
 397       integer_round_nearest                19                30
 398                        real            18.701            30.299
 399 </pre>
 400 </div>
 401 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 402 <td align="left"></td>
 403 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 404       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 405       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 406       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 407       Daryle Walker and Xiaogang Zhang<p>
 408         Distributed under the Boost Software License, Version 1.0. (See accompanying
 409         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>)
 410       </p>
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