| <html> |
| <head> |
| <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> |
| <title>Estimating Sample Sizes for a Binomial Distribution.</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="../binom_eg.html" title="Binomial Distribution Examples"> |
| <link rel="prev" href="binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for a Binomial Distribution"> |
| <link rel="next" href="../neg_binom_eg.html" title="Negative Binomial Distribution Examples"> |
| </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="binom_conf.html"><img src="../../../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../binom_eg.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="../neg_binom_eg.html"><img src="../../../../../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h6 class="title"> |
| <a name="math_toolkit.dist.stat_tut.weg.binom_eg.binom_size_eg"></a><a class="link" href="binom_size_eg.html" title="Estimating Sample Sizes for a Binomial Distribution."> |
| Estimating Sample Sizes for a Binomial Distribution.</a> |
| </h6></div></div></div> |
| <p> |
| Imagine you have a critical component that you know will fail in 1 |
| in N "uses" (for some suitable definition of "use"). |
| You may want to schedule routine replacement of the component so that |
| its chance of failure between routine replacements is less than P%. |
| If the failures follow a binomial distribution (each time the component |
| is "used" it either fails or does not) then the static member |
| function <code class="computeroutput"><span class="identifier">binomial_distibution</span><span class="special"><>::</span><span class="identifier">find_maximum_number_of_trials</span></code> |
| can be used to estimate the maximum number of "uses" of that |
| component for some acceptable risk level <span class="emphasis"><em>alpha</em></span>. |
| </p> |
| <p> |
| The example program <a href="../../../../../../../../example/binomial_sample_sizes.cpp" target="_top">binomial_sample_sizes.cpp</a> |
| demonstrates its usage. It centres on a routine that prints out a table |
| of maximum sample sizes for various probability thresholds: |
| </p> |
| <pre class="programlisting"><span class="keyword">void</span> <span class="identifier">find_max_sample_size</span><span class="special">(</span> |
| <span class="keyword">double</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// success ratio. |
| </span> <span class="keyword">unsigned</span> <span class="identifier">successes</span><span class="special">)</span> <span class="comment">// Total number of observed successes permitted. |
| </span><span class="special">{</span> |
| </pre> |
| <p> |
| The routine then declares a table of probability thresholds: these |
| are the maximum acceptable probability that <span class="emphasis"><em>successes</em></span> |
| or fewer events will be observed. In our example, <span class="emphasis"><em>successes</em></span> |
| will be always zero, since we want no component failures, but in other |
| situations non-zero values may well make sense. |
| </p> |
| <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.25</span><span class="special">,</span> <span class="number">0.1</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="number">0.01</span><span class="special">,</span> <span class="number">0.001</span><span class="special">,</span> <span class="number">0.0001</span><span class="special">,</span> <span class="number">0.00001</span> <span class="special">};</span> |
| </pre> |
| <p> |
| Much of the rest of the program is pretty-printing, the important part |
| is in the calculation of maximum number of permitted trials for each |
| value of alpha: |
| </p> |
| <pre class="programlisting"><span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special"><</span> <span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">)/</span><span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">[</span><span class="number">0</span><span class="special">]);</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span> |
| <span class="special">{</span> |
| <span class="comment">// Confidence value: |
| </span> <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">fixed</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</span> <span class="number">100</span> <span class="special">*</span> <span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]);</span> |
| <span class="comment">// calculate trials: |
| </span> <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">binomial</span><span class="special">::</span><span class="identifier">find_maximum_number_of_trials</span><span class="special">(</span> |
| <span class="identifier">successes</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]);</span> |
| <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">floor</span><span class="special">(</span><span class="identifier">t</span><span class="special">);</span> |
| <span class="comment">// Print Trials: |
| </span> <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">fixed</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</span> <span class="identifier">t</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="special">}</span> |
| </pre> |
| <p> |
| Note that since we're calculating the maximum number of trials permitted, |
| we'll err on the safe side and take the floor of the result. Had we |
| been calculating the <span class="emphasis"><em>minimum</em></span> number of trials |
| required to observe a certain number of <span class="emphasis"><em>successes</em></span> |
| using <code class="computeroutput"><span class="identifier">find_minimum_number_of_trials</span></code> |
| we would have taken the ceiling instead. |
| </p> |
| <p> |
| We'll finish off by looking at some sample output, firstly for a 1 |
| in 1000 chance of component failure with each use: |
| </p> |
| <pre class="programlisting">________________________ |
| Maximum Number of Trials |
| ________________________ |
| |
| Success ratio = 0.001 |
| Maximum Number of "successes" permitted = 0 |
| |
| |
| ____________________________ |
| Confidence Max Number |
| Value (%) Of Trials |
| ____________________________ |
| 50.000 692 |
| 75.000 287 |
| 90.000 105 |
| 95.000 51 |
| 99.000 10 |
| 99.900 0 |
| 99.990 0 |
| 99.999 0 |
| </pre> |
| <p> |
| So 51 "uses" of the component would yield a 95% chance that |
| no component failures would be observed. |
| </p> |
| <p> |
| Compare that with a 1 in 1 million chance of component failure: |
| </p> |
| <pre class="programlisting">________________________ |
| Maximum Number of Trials |
| ________________________ |
| |
| Success ratio = 0.0000010 |
| Maximum Number of "successes" permitted = 0 |
| |
| |
| ____________________________ |
| Confidence Max Number |
| Value (%) Of Trials |
| ____________________________ |
| 50.000 693146 |
| 75.000 287681 |
| 90.000 105360 |
| 95.000 51293 |
| 99.000 10050 |
| 99.900 1000 |
| 99.990 100 |
| 99.999 10 |
| </pre> |
| <p> |
| In this case, even 1000 uses of the component would still yield a less |
| than 1 in 1000 chance of observing a component failure (i.e. a 99.9% |
| chance of no failure). |
| </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, 2010 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="binom_conf.html"><img src="../../../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../binom_eg.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="../neg_binom_eg.html"><img src="../../../../../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| </body> |
| </html> |
