<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Understanding Quantiles of Discrete Distributions</title>
<link rel="stylesheet" href="../../../../../../../../doc/src/boostbook.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.74.0">
<link rel="home" href="../../../index.html" title="Math Toolkit">
<link rel="up" href="../pol_tutorial.html" title="Policy Tutorial">
<link rel="prev" href="user_def_err_pol.html" title="Calling User Defined Error Handlers">
<link rel="next" href="../pol_ref.html" title="Policy Reference">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
<table cellpadding="2" width="100%"><tr>
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../boost.png"></td>
<td align="center"><a href="../../../../../../../../index.html">Home</a></td>
<td align="center"><a href="../../../../../../../../libs/libraries.htm">Libraries</a></td>
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
<td align="center"><a href="../../../../../../../../more/index.htm">More</a></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="user_def_err_pol.html"><img src="../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../pol_tutorial.html"><img src="../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../pol_ref.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
<div class="section" lang="en">
<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.policy.pol_tutorial.understand_dis_quant"></a><a class="link" href="understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">
Understanding Quantiles of Discrete Distributions</a>
</h4></div></div></div>
<p>
Discrete distributions present us with a problem when calculating the quantile:
we are starting from a continuous real-valued variable - the probability
- but the result (the value of the random variable) should really be discrete.
</p>
<p>
Consider for example a Binomial distribution, with a sample size of 50,
and a success fraction of 0.5. There are a variety of ways we can plot
a discrete distribution, but if we plot the PDF as a step-function then
it looks something like this:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/binomial_pdf.png" alt="binomial_pdf"></span>
</p>
<p>
Now lets suppose that the user asks for a the quantile that corresponds
to a probability of 0.05, if we zoom in on the CDF for that region here's
what we see:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/binomial_quantile_1.png" alt="binomial_quantile_1"></span>
</p>
<p>
As can be seen there is no random variable that corresponds to a probability
of exactly 0.05, so we're left with two choices as shown in the figure:
</p>
<div class="itemizedlist"><ul type="disc">
<li>
We could round the result down to 18.
</li>
<li>
We could round the result up to 19.
</li>
</ul></div>
<p>
In fact there's actually a third choice as well: we could "pretend"
that the distribution was continuous and return a real valued result: in
this case we would calculate a result of approximately 18.701 (this accurately
reflects the fact that the result is nearer to 19 than 18).
</p>
<p>
By using policies we can offer any of the above as options, but that still
leaves the question: <span class="emphasis"><em>What is actually the right thing to do?</em></span>
</p>
<p>
And in particular: <span class="emphasis"><em>What policy should we use by default?</em></span>
</p>
<p>
In coming to an answer we should realise that:
</p>
<div class="itemizedlist"><ul type="disc">
<li>
Calculating an integer result is often much faster than calculating
a real-valued result: in fact in our tests it was up to 20 times faster.
</li>
<li>
Normally people calculate quantiles so that they can perform a test
of some kind: <span class="emphasis"><em>"If the random variable is less than N
then we can reject our null-hypothesis with 90% confidence."</em></span>
</li>
</ul></div>
<p>
So there is a genuine benefit to calculating an integer result as well
as it being "the right thing to do" from a philosophical point
of view. What's more if someone asks for a quantile at 0.05, then we can
normally assume that they are asking for <span class="emphasis"><em><span class="bold"><strong>at
least</strong></span> 95% of the probability to the right of the value chosen,
and <span class="bold"><strong>no more than</strong></span> 5% of the probability
to the left of the value chosen.</em></span>
</p>
<p>
In the above binomial example we would therefore round the result down
to 18.
</p>
<p>
The converse applies to upper-quantiles: If the probability is greater
than 0.5 we would want to round the quantile up, <span class="emphasis"><em>so that <span class="bold"><strong>at least</strong></span> the requested probability is to the left
of the value returned, and <span class="bold"><strong>no more than</strong></span>
1 - the requested probability is to the right of the value returned.</em></span>
</p>
<p>
Likewise for two-sided intervals, we would round lower quantiles down,
and upper quantiles up. This ensures that we have <span class="emphasis"><em>at least the
requested probability in the central region</em></span> and <span class="emphasis"><em>no
more than 1 minus the requested probability in the tail areas.</em></span>
</p>
<p>
For example, taking our 50 sample binomial distribution with a success
fraction of 0.5, if we wanted a two sided 90% confidence interval, then
we would ask for the 0.05 and 0.95 quantiles with the results <span class="emphasis"><em>rounded
outwards</em></span> so that <span class="emphasis"><em>at least 90% of the probability</em></span>
is in the central area:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/binomial_pdf_3.png" alt="binomial_pdf_3"></span>
</p>
<p>
So far so good, but there is in fact a trap waiting for the unwary here:
</p>
<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>
</pre>
<p>
returns 18 as the result, which is what we would expect from the graph
above, and indeed there is no x greater than 18 for which:
</p>
<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"><=</span> <span class="number">0.05</span><span class="special">;</span>
</pre>
<p>
However:
</p>
<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>
</pre>
<p>
returns 31, and indeed while there is no x less than 31 for which:
</p>
<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">>=</span> <span class="number">0.95</span><span class="special">;</span>
</pre>
<p>
We might naively expect that for this symmetrical distribution the result
would be 32 (since 32 = 50 - 18), but we need to remember that the cdf
of the binomial is <span class="emphasis"><em>inclusive</em></span> of the random variable.
So while the left tail area <span class="emphasis"><em>includes</em></span> the quantile
returned, the right tail area always excludes an upper quantile value:
since that "belongs" to the central area.
</p>
<p>
Look at the graph above to see what's going on here: the lower quantile
of 18 belongs to the left tail, so any value <= 18 is in the left tail.
The upper quantile of 31 on the other hand belongs to the central area,
so the tail area actually starts at 32, so any value > 31 is in the
right tail.
</p>
<p>
Therefore if U and L are the upper and lower quantiles respectively, then
a random variable X is in the tail area - where we would reject the null
hypothesis if:
</p>
<pre class="programlisting"><span class="identifier">X</span> <span class="special"><=</span> <span class="identifier">L</span> <span class="special">||</span> <span class="identifier">X</span> <span class="special">></span> <span class="identifier">U</span>
</pre>
<p>
And the a variable X is inside the central region if:
</p>
<pre class="programlisting"><span class="identifier">L</span> <span class="special"><</span> <span class="identifier">X</span> <span class="special"><=</span> <span class="identifier">U</span>
</pre>
<p>
The moral here is to <span class="emphasis"><em>always be very careful with your comparisons
when dealing with a discrete distribution</em></span>, and if in doubt,
<span class="emphasis"><em>base your comparisons on CDF's instead</em></span>.
</p>
<a name="math_toolkit.policy.pol_tutorial.understand_dis_quant.other_rounding_policies_are_available"></a><h6>
<a name="id1258022"></a>
<a class="link" href="understand_dis_quant.html#math_toolkit.policy.pol_tutorial.understand_dis_quant.other_rounding_policies_are_available">Other
Rounding Policies are Available</a>
</h6>
<p>
As you would expect from a section on policies, you won't be surprised
to know that other rounding options are available:
</p>
<div class="variablelist">
<p class="title"><b></b></p>
<dl>
<dt><span class="term">integer_round_outwards</span></dt>
<dd>
<p>
This is the default policy as described above: lower quantiles are
rounded down (probability < 0.5), and upper quantiles (probability
> 0.5) are rounded up.
</p>
<p>
This gives <span class="emphasis"><em>no more than</em></span> the requested probability
in the tails, and <span class="emphasis"><em>at least</em></span> the requested probability
in the central area.
</p>
</dd>
<dt><span class="term">integer_round_inwards</span></dt>
<dd>
<p>
This is the exact opposite of the default policy: lower quantiles
are rounded up (probability < 0.5), and upper quantiles (probability
> 0.5) are rounded down.
</p>
<p>
This gives <span class="emphasis"><em>at least</em></span> the requested probability
in the tails, and <span class="emphasis"><em>no more than</em></span> the requested
probability in the central area.
</p>
</dd>
<dt><span class="term">integer_round_down</span></dt>
<dd><p>
This policy will always round the result down no matter whether it
is an upper or lower quantile
</p></dd>
<dt><span class="term">integer_round_up</span></dt>
<dd><p>
This policy will always round the result up no matter whether it
is an upper or lower quantile
</p></dd>
<dt><span class="term">integer_round_nearest</span></dt>
<dd><p>
This policy will always round the result to the nearest integer no
matter whether it is an upper or lower quantile
</p></dd>
<dt><span class="term">real</span></dt>
<dd><p>
This policy will return a real valued result for the quantile of
a discrete distribution: this is generally much slower than finding
an integer result but does allow for more sophisticated rounding
policies.
</p></dd>
</dl>
</div>
<p>
</p>
<p>
To understand how the rounding policies for the discrete distributions
can be used, we'll use the 50-sample binomial distribution with a success
fraction of 0.5 once again, and calculate all the possible quantiles
at 0.05 and 0.95.
</p>
<p>
</p>
<p>
Begin by including the needed headers:
</p>
<p>
</p>
<p>
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span>
<span class="preprocessor">#include</span> <span class="special"><</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">></span></pre>
<p>
</p>
<p>
</p>
<p>
Next we'll bring the needed declarations into scope, and define distribution
types for all the available rounding policies:
</p>
<p>
</p>
<p>
</p>
<pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</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="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special"><</span>
<span class="keyword">double</span><span class="special">,</span>
<span class="identifier">policy</span><span class="special"><</span><span class="identifier">discrete_quantile</span><span class="special"><</span><span class="identifier">integer_round_outwards</span><span class="special">></span> <span class="special">></span> <span class="special">></span>
<span class="identifier">binom_round_outwards</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special"><</span>
<span class="keyword">double</span><span class="special">,</span>
<span class="identifier">policy</span><span class="special"><</span><span class="identifier">discrete_quantile</span><span class="special"><</span><span class="identifier">integer_round_inwards</span><span class="special">></span> <span class="special">></span> <span class="special">></span>
<span class="identifier">binom_round_inwards</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special"><</span>
<span class="keyword">double</span><span class="special">,</span>
<span class="identifier">policy</span><span class="special"><</span><span class="identifier">discrete_quantile</span><span class="special"><</span><span class="identifier">integer_round_down</span><span class="special">></span> <span class="special">></span> <span class="special">></span>
<span class="identifier">binom_round_down</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special"><</span>
<span class="keyword">double</span><span class="special">,</span>
<span class="identifier">policy</span><span class="special"><</span><span class="identifier">discrete_quantile</span><span class="special"><</span><span class="identifier">integer_round_up</span><span class="special">></span> <span class="special">></span> <span class="special">></span>
<span class="identifier">binom_round_up</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special"><</span>
<span class="keyword">double</span><span class="special">,</span>
<span class="identifier">policy</span><span class="special"><</span><span class="identifier">discrete_quantile</span><span class="special"><</span><span class="identifier">integer_round_nearest</span><span class="special">></span> <span class="special">></span> <span class="special">></span>
<span class="identifier">binom_round_nearest</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special"><</span>
<span class="keyword">double</span><span class="special">,</span>
<span class="identifier">policy</span><span class="special"><</span><span class="identifier">discrete_quantile</span><span class="special"><</span><span class="identifier">real</span><span class="special">></span> <span class="special">></span> <span class="special">></span>
<span class="identifier">binom_real_quantile</span><span class="special">;</span></pre>
<p>
</p>
<p>
</p>
<p>
Now let's set to work calling those quantiles:
</p>
<p>
</p>
<p>
</p>
<pre class="programlisting"><span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
<span class="special">{</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span>
<span class="string">"Testing rounding policies for a 50 sample binomial distribution,\n"</span>
<span class="string">"with a success fraction of 0.5.\n\n"</span>
<span class="string">"Lower quantiles are calculated at p = 0.05\n\n"</span>
<span class="string">"Upper quantiles at p = 0.95.\n\n"</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span>
<span class="special"><<</span> <span class="string">"Policy"</span><span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span>
<span class="special"><<</span> <span class="string">"Lower Quantile"</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span>
<span class="special"><<</span> <span class="string">"Upper Quantile"</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Test integer_round_outwards:
</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span>
<span class="special"><<</span> <span class="string">"integer_round_outwards"</span>
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span>
<span class="special"><<</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>
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span>
<span class="special"><<</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>
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Test integer_round_inwards:
</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span>
<span class="special"><<</span> <span class="string">"integer_round_inwards"</span>
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span>
<span class="special"><<</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>
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span>
<span class="special"><<</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>
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Test integer_round_down:
</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span>
<span class="special"><<</span> <span class="string">"integer_round_down"</span>
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span>
<span class="special"><<</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>
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span>
<span class="special"><<</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>
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Test integer_round_up:
</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span>
<span class="special"><<</span> <span class="string">"integer_round_up"</span>
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span>
<span class="special"><<</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>
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span>
<span class="special"><<</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>
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Test integer_round_nearest:
</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span>
<span class="special"><<</span> <span class="string">"integer_round_nearest"</span>
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span>
<span class="special"><<</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>
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span>
<span class="special"><<</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>
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="comment">// Test real:
</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span>
<span class="special"><<</span> <span class="string">"real"</span>
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span>
<span class="special"><<</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>
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span>
<span class="special"><<</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>
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span></pre>
<p>
</p>
<p>
</p>
<p>
Which produces the program output:
</p>
<p>
</p>
<pre class="programlisting">Testing rounding policies for a 50 sample binomial distribution,
with a success fraction of 0.5.
Lower quantiles are calculated at p = 0.05
Upper quantiles at p = 0.95.
Testing rounding policies for a 50 sample binomial distribution,
with a success fraction of 0.5.
Lower quantiles are calculated at p = 0.05
Upper quantiles at p = 0.95.
Policy Lower Quantile Upper Quantile
integer_round_outwards 18 31
integer_round_inwards 19 30
integer_round_down 18 30
integer_round_up 19 31
integer_round_nearest 19 30
real 18.701 30.299
</pre>
<p>
</p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009 John Maddock, Paul A. Bristow,
Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani
and Thijs van den Berg<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
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>)
</p>
</div></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="user_def_err_pol.html"><img src="../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../pol_tutorial.html"><img src="../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../pol_ref.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
</body>
</html>
