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<p><a href="https://en.wikipedia.org/wiki/Binomial_distribution" target="_blank">Binomial distribution</a> example
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;cmath&gt;</code><br />
<code>#include &lt;iostream&gt;</code><br />
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Include dependency graph for binomial_dist.cpp:</div>
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<p><a href="../../d6/db0/binomial__dist_8cpp_source.html">Go to the source code of this file.</a></p>
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Functions</h2></td></tr>
<tr class="memitem:a4416a7bc7fa87201883c54cdc4c82813" id="r_a4416a7bc7fa87201883c54cdc4c82813"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a4416a7bc7fa87201883c54cdc4c82813">binomial_expected</a> (double n, double p)</td></tr>
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<tr class="memitem:acd4dd4558031e4c5d045c801f73d8861" id="r_acd4dd4558031e4c5d045c801f73d8861"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#acd4dd4558031e4c5d045c801f73d8861">binomial_variance</a> (double n, double p)</td></tr>
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<tr class="memitem:af09e51f513cee647d41192ab0a872cdc" id="r_af09e51f513cee647d41192ab0a872cdc"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#af09e51f513cee647d41192ab0a872cdc">binomial_standard_deviation</a> (double n, double p)</td></tr>
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<tr class="memitem:a78d36635232e54b5d71fcbf1eac9a49a" id="r_a78d36635232e54b5d71fcbf1eac9a49a"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a78d36635232e54b5d71fcbf1eac9a49a">nCr</a> (double n, double r)</td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p><a href="https://en.wikipedia.org/wiki/Binomial_distribution" target="_blank">Binomial distribution</a> example </p>
<p>The binomial distribution models the number of successes in a sequence of n independent events</p>
<p>Summary of variables used:</p><ul>
<li>n : number of trials</li>
<li>p : probability of success</li>
<li>x : desired successes </li>
</ul>
<p class="definition">Definition in file <a class="el" href="../../d6/db0/binomial__dist_8cpp_source.html">binomial_dist.cpp</a>.</p>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="a4416a7bc7fa87201883c54cdc4c82813" name="a4416a7bc7fa87201883c54cdc4c82813"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a4416a7bc7fa87201883c54cdc4c82813">&#9670;&#160;</a></span>binomial_expected()</h2>
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<td class="memname">double binomial_expected </td>
<td>(</td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>n</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>p</em></span>&#160;)</td>
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</table>
</div><div class="memdoc">
<p>finds the expected value of a binomial distribution </p><dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">n</td><td></td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">p</td><td></td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\(\mu=np\) </dd></dl>
<p class="definition">Definition at line <a class="el" href="../../d6/db0/binomial__dist_8cpp_source.html#l00022">22</a> of file <a class="el" href="../../d6/db0/binomial__dist_8cpp_source.html">binomial_dist.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 22</span>{ <span class="keywordflow">return</span> n * p; }</div>
</div><!-- fragment -->
</div>
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<a id="a76ed6ce71415fb400b65f0656cef3d25" name="a76ed6ce71415fb400b65f0656cef3d25"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a76ed6ce71415fb400b65f0656cef3d25">&#9670;&#160;</a></span>binomial_range_successes()</h2>
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<td class="memname">double binomial_range_successes </td>
<td>(</td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>n</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>p</em></span>, </td>
</tr>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>lower_bound</em></span>, </td>
</tr>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>upper_bound</em></span>&#160;)</td>
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</div><div class="memdoc">
<p>calculates the probability of a result within a range (inclusive, inclusive) </p><dl class="section return"><dt>Returns</dt><dd>\(\displaystyle \left.P(n,p)\right|_{x_0}^{x_1} =
\sum_{i=x_0}^{x_1} P(i)
=\sum_{i=x_0}^{x_1} {n\choose i} p^i (1-p)^{n-i}\) </dd></dl>
<p class="definition">Definition at line <a class="el" href="../../d6/db0/binomial__dist_8cpp_source.html#l00074">74</a> of file <a class="el" href="../../d6/db0/binomial__dist_8cpp_source.html">binomial_dist.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 75</span> {</div>
<div class="line"><span class="lineno"> 76</span> <span class="keywordtype">double</span> <a class="code hl_namespace" href="../../d4/ded/namespaceprobability.html">probability</a> = 0;</div>
<div class="line"><span class="lineno"> 77</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = lower_bound; i &lt;= upper_bound; i++) {</div>
<div class="line"><span class="lineno"> 78</span> <a class="code hl_namespace" href="../../d4/ded/namespaceprobability.html">probability</a> += <a class="code hl_function" href="#a78d36635232e54b5d71fcbf1eac9a49a">nCr</a>(n, i) * std::pow(p, i) * std::pow(1 - p, n - i);</div>
<div class="line"><span class="lineno"> 79</span> }</div>
<div class="line"><span class="lineno"> 80</span> <span class="keywordflow">return</span> <a class="code hl_namespace" href="../../d4/ded/namespaceprobability.html">probability</a>;</div>
<div class="line"><span class="lineno"> 81</span>}</div>
<div class="ttc" id="abinomial__dist_8cpp_html_a78d36635232e54b5d71fcbf1eac9a49a"><div class="ttname"><a href="#a78d36635232e54b5d71fcbf1eac9a49a">nCr</a></div><div class="ttdeci">double nCr(double n, double r)</div><div class="ttdef"><b>Definition</b> <a href="../../d6/db0/binomial__dist_8cpp_source.html#l00047">binomial_dist.cpp:47</a></div></div>
<div class="ttc" id="anamespaceprobability_html"><div class="ttname"><a href="../../d4/ded/namespaceprobability.html">probability</a></div><div class="ttdoc">Probability algorithms.</div></div>
</div><!-- fragment -->
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</div>
<a id="af09e51f513cee647d41192ab0a872cdc" name="af09e51f513cee647d41192ab0a872cdc"></a>
<h2 class="memtitle"><span class="permalink"><a href="#af09e51f513cee647d41192ab0a872cdc">&#9670;&#160;</a></span>binomial_standard_deviation()</h2>
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<td class="memname">double binomial_standard_deviation </td>
<td>(</td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>n</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>p</em></span>&#160;)</td>
</tr>
</table>
</div><div class="memdoc">
<p>finds the standard deviation of the binomial distribution </p><dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">n</td><td></td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">p</td><td></td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\(\sigma = \sqrt{\sigma^2} = \sqrt{n\cdot p\cdot (1-p)}\) </dd></dl>
<p class="definition">Definition at line <a class="el" href="../../d6/db0/binomial__dist_8cpp_source.html#l00036">36</a> of file <a class="el" href="../../d6/db0/binomial__dist_8cpp_source.html">binomial_dist.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 36</span> {</div>
<div class="line"><span class="lineno"> 37</span> <span class="keywordflow">return</span> std::sqrt(<a class="code hl_function" href="#acd4dd4558031e4c5d045c801f73d8861">binomial_variance</a>(n, p));</div>
<div class="line"><span class="lineno"> 38</span>}</div>
<div class="ttc" id="abinomial__dist_8cpp_html_acd4dd4558031e4c5d045c801f73d8861"><div class="ttname"><a href="#acd4dd4558031e4c5d045c801f73d8861">binomial_variance</a></div><div class="ttdeci">double binomial_variance(double n, double p)</div><div class="ttdef"><b>Definition</b> <a href="../../d6/db0/binomial__dist_8cpp_source.html#l00029">binomial_dist.cpp:29</a></div></div>
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<a id="acd4dd4558031e4c5d045c801f73d8861" name="acd4dd4558031e4c5d045c801f73d8861"></a>
<h2 class="memtitle"><span class="permalink"><a href="#acd4dd4558031e4c5d045c801f73d8861">&#9670;&#160;</a></span>binomial_variance()</h2>
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<td class="memname">double binomial_variance </td>
<td>(</td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>n</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>p</em></span>&#160;)</td>
</tr>
</table>
</div><div class="memdoc">
<p>finds the variance of the binomial distribution </p><dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">n</td><td></td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">p</td><td></td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\(\sigma^2 = n\cdot p\cdot (1-p)\) </dd></dl>
<p class="definition">Definition at line <a class="el" href="../../d6/db0/binomial__dist_8cpp_source.html#l00029">29</a> of file <a class="el" href="../../d6/db0/binomial__dist_8cpp_source.html">binomial_dist.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 29</span>{ <span class="keywordflow">return</span> n * p * (1 - p); }</div>
</div><!-- fragment -->
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<a id="a19ae0a6a2bd200fd1eb0e31b2bf4cc76" name="a19ae0a6a2bd200fd1eb0e31b2bf4cc76"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a19ae0a6a2bd200fd1eb0e31b2bf4cc76">&#9670;&#160;</a></span>binomial_x_successes()</h2>
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<td class="memname">double binomial_x_successes </td>
<td>(</td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>n</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>p</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>x</em></span>&#160;)</td>
</tr>
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</div><div class="memdoc">
<p>calculates the probability of exactly x successes </p><dl class="section return"><dt>Returns</dt><dd>\(\displaystyle P(n,p,x) = {n\choose x} p^x (1-p)^{n-x}\) </dd></dl>
<p class="definition">Definition at line <a class="el" href="../../d6/db0/binomial__dist_8cpp_source.html#l00065">65</a> of file <a class="el" href="../../d6/db0/binomial__dist_8cpp_source.html">binomial_dist.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 65</span> {</div>
<div class="line"><span class="lineno"> 66</span> <span class="keywordflow">return</span> <a class="code hl_function" href="#a78d36635232e54b5d71fcbf1eac9a49a">nCr</a>(n, x) * std::pow(p, x) * std::pow(1 - p, n - x);</div>
<div class="line"><span class="lineno"> 67</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#ae66f6b31b5ad750f1fe042a706a4e3d4">&#9670;&#160;</a></span>main()</h2>
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<td class="memname">int main </td>
<td>(</td>
<td class="paramtype">void</td> <td class="paramname"><span class="paramname"><em></em></span></td><td>)</td>
<td></td>
</tr>
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<p>main function </p>
<p class="definition">Definition at line <a class="el" href="../../d6/db0/binomial__dist_8cpp_source.html#l00084">84</a> of file <a class="el" href="../../d6/db0/binomial__dist_8cpp_source.html">binomial_dist.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 84</span> {</div>
<div class="line"><span class="lineno"> 85</span> std::cout &lt;&lt; <span class="stringliteral">&quot;expected value : &quot;</span> &lt;&lt; <a class="code hl_function" href="#a4416a7bc7fa87201883c54cdc4c82813">binomial_expected</a>(100, 0.5)</div>
<div class="line"><span class="lineno"> 86</span> &lt;&lt; std::endl;</div>
<div class="line"><span class="lineno"> 87</span> </div>
<div class="line"><span class="lineno"> 88</span> std::cout &lt;&lt; <span class="stringliteral">&quot;variance : &quot;</span> &lt;&lt; <a class="code hl_function" href="#acd4dd4558031e4c5d045c801f73d8861">binomial_variance</a>(100, 0.5) &lt;&lt; std::endl;</div>
<div class="line"><span class="lineno"> 89</span> </div>
<div class="line"><span class="lineno"> 90</span> std::cout &lt;&lt; <span class="stringliteral">&quot;standard deviation : &quot;</span></div>
<div class="line"><span class="lineno"> 91</span> &lt;&lt; <a class="code hl_function" href="#af09e51f513cee647d41192ab0a872cdc">binomial_standard_deviation</a>(100, 0.5) &lt;&lt; std::endl;</div>
<div class="line"><span class="lineno"> 92</span> </div>
<div class="line"><span class="lineno"> 93</span> std::cout &lt;&lt; <span class="stringliteral">&quot;exactly 30 successes : &quot;</span> &lt;&lt; <a class="code hl_function" href="#a19ae0a6a2bd200fd1eb0e31b2bf4cc76">binomial_x_successes</a>(100, 0.5, 30)</div>
<div class="line"><span class="lineno"> 94</span> &lt;&lt; std::endl;</div>
<div class="line"><span class="lineno"> 95</span> </div>
<div class="line"><span class="lineno"> 96</span> std::cout &lt;&lt; <span class="stringliteral">&quot;45 or more successes : &quot;</span></div>
<div class="line"><span class="lineno"> 97</span> &lt;&lt; <a class="code hl_function" href="#a76ed6ce71415fb400b65f0656cef3d25">binomial_range_successes</a>(100, 0.5, 45, 100) &lt;&lt; std::endl;</div>
<div class="line"><span class="lineno"> 98</span> </div>
<div class="line"><span class="lineno"> 99</span> <span class="keywordflow">return</span> 0;</div>
<div class="line"><span class="lineno"> 100</span>}</div>
<div class="ttc" id="abinomial__dist_8cpp_html_a19ae0a6a2bd200fd1eb0e31b2bf4cc76"><div class="ttname"><a href="#a19ae0a6a2bd200fd1eb0e31b2bf4cc76">binomial_x_successes</a></div><div class="ttdeci">double binomial_x_successes(double n, double p, double x)</div><div class="ttdef"><b>Definition</b> <a href="../../d6/db0/binomial__dist_8cpp_source.html#l00065">binomial_dist.cpp:65</a></div></div>
<div class="ttc" id="abinomial__dist_8cpp_html_a4416a7bc7fa87201883c54cdc4c82813"><div class="ttname"><a href="#a4416a7bc7fa87201883c54cdc4c82813">binomial_expected</a></div><div class="ttdeci">double binomial_expected(double n, double p)</div><div class="ttdef"><b>Definition</b> <a href="../../d6/db0/binomial__dist_8cpp_source.html#l00022">binomial_dist.cpp:22</a></div></div>
<div class="ttc" id="abinomial__dist_8cpp_html_a76ed6ce71415fb400b65f0656cef3d25"><div class="ttname"><a href="#a76ed6ce71415fb400b65f0656cef3d25">binomial_range_successes</a></div><div class="ttdeci">double binomial_range_successes(double n, double p, double lower_bound, double upper_bound)</div><div class="ttdef"><b>Definition</b> <a href="../../d6/db0/binomial__dist_8cpp_source.html#l00074">binomial_dist.cpp:74</a></div></div>
<div class="ttc" id="abinomial__dist_8cpp_html_af09e51f513cee647d41192ab0a872cdc"><div class="ttname"><a href="#af09e51f513cee647d41192ab0a872cdc">binomial_standard_deviation</a></div><div class="ttdeci">double binomial_standard_deviation(double n, double p)</div><div class="ttdef"><b>Definition</b> <a href="../../d6/db0/binomial__dist_8cpp_source.html#l00036">binomial_dist.cpp:36</a></div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#a78d36635232e54b5d71fcbf1eac9a49a">&#9670;&#160;</a></span>nCr()</h2>
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<td class="memname">double nCr </td>
<td>(</td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>n</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>r</em></span>&#160;)</td>
</tr>
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<p>Computes n choose r </p><dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">n</td><td></td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">r</td><td></td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\(\displaystyle {n\choose r} =
\frac{n!}{r!(n-r)!} = \frac{n\times(n-1)\times(n-2)\times\cdots(n-r)}{r!}
\) </dd></dl>
<p class="definition">Definition at line <a class="el" href="../../d6/db0/binomial__dist_8cpp_source.html#l00047">47</a> of file <a class="el" href="../../d6/db0/binomial__dist_8cpp_source.html">binomial_dist.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 47</span> {</div>
<div class="line"><span class="lineno"> 48</span> <span class="keywordtype">double</span> numerator = n;</div>
<div class="line"><span class="lineno"> 49</span> <span class="keywordtype">double</span> denominator = r;</div>
<div class="line"><span class="lineno"> 50</span> </div>
<div class="line"><span class="lineno"> 51</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = n - 1; i &gt;= ((n - r) + 1); i--) {</div>
<div class="line"><span class="lineno"> 52</span> numerator *= i;</div>
<div class="line"><span class="lineno"> 53</span> }</div>
<div class="line"><span class="lineno"> 54</span> </div>
<div class="line"><span class="lineno"> 55</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 1; i &lt; r; i++) {</div>
<div class="line"><span class="lineno"> 56</span> denominator *= i;</div>
<div class="line"><span class="lineno"> 57</span> }</div>
<div class="line"><span class="lineno"> 58</span> </div>
<div class="line"><span class="lineno"> 59</span> <span class="keywordflow">return</span> numerator / denominator;</div>
<div class="line"><span class="lineno"> 60</span>}</div>
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