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<p><a href="https://en.wikipedia.org/wiki/Binomial_distribution">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 />
</div><div class="textblock"><div class="dynheader">
Include dependency graph for binomial_dist.cpp:</div>
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Functions</h2></td></tr>
<tr class="memitem:a4416a7bc7fa87201883c54cdc4c82813"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/db0/binomial__dist_8cpp.html#a4416a7bc7fa87201883c54cdc4c82813">binomial_expected</a> (double n, double p)</td></tr>
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<tr class="memitem:acd4dd4558031e4c5d045c801f73d8861"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/db0/binomial__dist_8cpp.html#acd4dd4558031e4c5d045c801f73d8861">binomial_variance</a> (double n, double p)</td></tr>
<tr class="separator:acd4dd4558031e4c5d045c801f73d8861"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:af09e51f513cee647d41192ab0a872cdc"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/db0/binomial__dist_8cpp.html#af09e51f513cee647d41192ab0a872cdc">binomial_standard_deviation</a> (double n, double p)</td></tr>
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<tr class="memitem:a78d36635232e54b5d71fcbf1eac9a49a"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/db0/binomial__dist_8cpp.html#a78d36635232e54b5d71fcbf1eac9a49a">nCr</a> (double n, double r)</td></tr>
<tr class="separator:a78d36635232e54b5d71fcbf1eac9a49a"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a19ae0a6a2bd200fd1eb0e31b2bf4cc76"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/db0/binomial__dist_8cpp.html#a19ae0a6a2bd200fd1eb0e31b2bf4cc76">binomial_x_successes</a> (double n, double p, double x)</td></tr>
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<tr class="memitem:a76ed6ce71415fb400b65f0656cef3d25"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/db0/binomial__dist_8cpp.html#a76ed6ce71415fb400b65f0656cef3d25">binomial_range_successes</a> (double n, double p, double lower_bound, double upper_bound)</td></tr>
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<tr class="memitem:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/db0/binomial__dist_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a> ()</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">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>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="a4416a7bc7fa87201883c54cdc4c82813"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a4416a7bc7fa87201883c54cdc4c82813">&#9670;&nbsp;</a></span>binomial_expected()</h2>
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<td class="memname">double binomial_expected </td>
<td>(</td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>n</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>p</em>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</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>
<div class="fragment"><div class="line"><a name="l00022"></a><span class="lineno"> 22</span>&#160;{ <span class="keywordflow">return</span> n * p; }</div>
</div><!-- fragment -->
</div>
</div>
<a id="a76ed6ce71415fb400b65f0656cef3d25"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a76ed6ce71415fb400b65f0656cef3d25">&#9670;&nbsp;</a></span>binomial_range_successes()</h2>
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<td class="memname">double binomial_range_successes </td>
<td>(</td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>n</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>p</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>lower_bound</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>upper_bound</em>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
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<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>
<div class="fragment"><div class="line"><a name="l00075"></a><span class="lineno"> 75</span>&#160; {</div>
<div class="line"><a name="l00076"></a><span class="lineno"> 76</span>&#160; <span class="keywordtype">double</span> probability = 0;</div>
<div class="line"><a name="l00077"></a><span class="lineno"> 77</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = lower_bound; i &lt;= <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/upper_bound.html">upper_bound</a>; i++) {</div>
<div class="line"><a name="l00078"></a><span class="lineno"> 78</span>&#160; probability += <a class="code" href="../../d6/db0/binomial__dist_8cpp.html#a78d36635232e54b5d71fcbf1eac9a49a">nCr</a>(n, i) * <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/pow.html">std::pow</a>(p, i) * <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/pow.html">std::pow</a>(1 - p, n - i);</div>
<div class="line"><a name="l00079"></a><span class="lineno"> 79</span>&#160; }</div>
<div class="line"><a name="l00080"></a><span class="lineno"> 80</span>&#160; <span class="keywordflow">return</span> probability;</div>
<div class="line"><a name="l00081"></a><span class="lineno"> 81</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#af09e51f513cee647d41192ab0a872cdc">&#9670;&nbsp;</a></span>binomial_standard_deviation()</h2>
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<td class="memname">double binomial_standard_deviation </td>
<td>(</td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>n</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>p</em>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></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>
<div class="fragment"><div class="line"><a name="l00036"></a><span class="lineno"> 36</span>&#160; {</div>
<div class="line"><a name="l00037"></a><span class="lineno"> 37</span>&#160; <span class="keywordflow">return</span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">std::sqrt</a>(<a class="code" href="../../d6/db0/binomial__dist_8cpp.html#acd4dd4558031e4c5d045c801f73d8861">binomial_variance</a>(n, p));</div>
<div class="line"><a name="l00038"></a><span class="lineno"> 38</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#acd4dd4558031e4c5d045c801f73d8861">&#9670;&nbsp;</a></span>binomial_variance()</h2>
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<td class="memname">double binomial_variance </td>
<td>(</td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>n</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>p</em>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></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>
<div class="fragment"><div class="line"><a name="l00029"></a><span class="lineno"> 29</span>&#160;{ <span class="keywordflow">return</span> n * p * (1 - p); }</div>
</div><!-- fragment -->
</div>
</div>
<a id="a19ae0a6a2bd200fd1eb0e31b2bf4cc76"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a19ae0a6a2bd200fd1eb0e31b2bf4cc76">&#9670;&nbsp;</a></span>binomial_x_successes()</h2>
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<td class="memname">double binomial_x_successes </td>
<td>(</td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>n</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>p</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>x</em>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></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>
<div class="fragment"><div class="line"><a name="l00065"></a><span class="lineno"> 65</span>&#160; {</div>
<div class="line"><a name="l00066"></a><span class="lineno"> 66</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../d6/db0/binomial__dist_8cpp.html#a78d36635232e54b5d71fcbf1eac9a49a">nCr</a>(n, x) * <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/pow.html">std::pow</a>(p, x) * <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/pow.html">std::pow</a>(1 - p, n - x);</div>
<div class="line"><a name="l00067"></a><span class="lineno"> 67</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#ae66f6b31b5ad750f1fe042a706a4e3d4">&#9670;&nbsp;</a></span>main()</h2>
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<td class="memname">int main </td>
<td>(</td>
<td class="paramname"></td><td>)</td>
<td></td>
</tr>
</table>
</div><div class="memdoc">
<p>main function </p>
<div class="fragment"><div class="line"><a name="l00084"></a><span class="lineno"> 84</span>&#160; {</div>
<div class="line"><a name="l00085"></a><span class="lineno"> 85</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;expected value : &quot;</span> &lt;&lt; <a class="code" href="../../d6/db0/binomial__dist_8cpp.html#a4416a7bc7fa87201883c54cdc4c82813">binomial_expected</a>(100, 0.5)</div>
<div class="line"><a name="l00086"></a><span class="lineno"> 86</span>&#160; &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><a name="l00087"></a><span class="lineno"> 87</span>&#160; </div>
<div class="line"><a name="l00088"></a><span class="lineno"> 88</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;variance : &quot;</span> &lt;&lt; <a class="code" href="../../d6/db0/binomial__dist_8cpp.html#acd4dd4558031e4c5d045c801f73d8861">binomial_variance</a>(100, 0.5) &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><a name="l00089"></a><span class="lineno"> 89</span>&#160; </div>
<div class="line"><a name="l00090"></a><span class="lineno"> 90</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;standard deviation : &quot;</span></div>
<div class="line"><a name="l00091"></a><span class="lineno"> 91</span>&#160; &lt;&lt; <a class="code" href="../../d6/db0/binomial__dist_8cpp.html#af09e51f513cee647d41192ab0a872cdc">binomial_standard_deviation</a>(100, 0.5) &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><a name="l00092"></a><span class="lineno"> 92</span>&#160; </div>
<div class="line"><a name="l00093"></a><span class="lineno"> 93</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;exactly 30 successes : &quot;</span> &lt;&lt; <a class="code" href="../../d6/db0/binomial__dist_8cpp.html#a19ae0a6a2bd200fd1eb0e31b2bf4cc76">binomial_x_successes</a>(100, 0.5, 30)</div>
<div class="line"><a name="l00094"></a><span class="lineno"> 94</span>&#160; &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><a name="l00095"></a><span class="lineno"> 95</span>&#160; </div>
<div class="line"><a name="l00096"></a><span class="lineno"> 96</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;45 or more successes : &quot;</span></div>
<div class="line"><a name="l00097"></a><span class="lineno"> 97</span>&#160; &lt;&lt; <a class="code" href="../../d6/db0/binomial__dist_8cpp.html#a76ed6ce71415fb400b65f0656cef3d25">binomial_range_successes</a>(100, 0.5, 45, 100) &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><a name="l00098"></a><span class="lineno"> 98</span>&#160; </div>
<div class="line"><a name="l00099"></a><span class="lineno"> 99</span>&#160; <span class="keywordflow">return</span> 0;</div>
<div class="line"><a name="l00100"></a><span class="lineno"> 100</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a78d36635232e54b5d71fcbf1eac9a49a">&#9670;&nbsp;</a></span>nCr()</h2>
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<td class="memname">double nCr </td>
<td>(</td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>n</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>r</em>&#160;</td>
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<tr>
<td></td>
<td>)</td>
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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>
<div class="fragment"><div class="line"><a name="l00047"></a><span class="lineno"> 47</span>&#160; {</div>
<div class="line"><a name="l00048"></a><span class="lineno"> 48</span>&#160; <span class="keywordtype">double</span> numerator = n;</div>
<div class="line"><a name="l00049"></a><span class="lineno"> 49</span>&#160; <span class="keywordtype">double</span> denominator = r;</div>
<div class="line"><a name="l00050"></a><span class="lineno"> 50</span>&#160; </div>
<div class="line"><a name="l00051"></a><span class="lineno"> 51</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = n - 1; i &gt;= ((n - r) + 1); i--) {</div>
<div class="line"><a name="l00052"></a><span class="lineno"> 52</span>&#160; numerator *= i;</div>
<div class="line"><a name="l00053"></a><span class="lineno"> 53</span>&#160; }</div>
<div class="line"><a name="l00054"></a><span class="lineno"> 54</span>&#160; </div>
<div class="line"><a name="l00055"></a><span class="lineno"> 55</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 1; i &lt; r; i++) {</div>
<div class="line"><a name="l00056"></a><span class="lineno"> 56</span>&#160; denominator *= i;</div>
<div class="line"><a name="l00057"></a><span class="lineno"> 57</span>&#160; }</div>
<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>&#160; </div>
<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>&#160; <span class="keywordflow">return</span> numerator / denominator;</div>
<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>&#160;}</div>
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<div class="ttc" id="asqrt_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">std::sqrt</a></div><div class="ttdeci">T sqrt(T... args)</div></div>
<div class="ttc" id="abasic_ostream_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a></div></div>
<div class="ttc" id="abinomial__dist_8cpp_html_a76ed6ce71415fb400b65f0656cef3d25"><div class="ttname"><a href="../../d6/db0/binomial__dist_8cpp.html#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> binomial_dist.cpp:74</div></div>
<div class="ttc" id="abinomial__dist_8cpp_html_af09e51f513cee647d41192ab0a872cdc"><div class="ttname"><a href="../../d6/db0/binomial__dist_8cpp.html#af09e51f513cee647d41192ab0a872cdc">binomial_standard_deviation</a></div><div class="ttdeci">double binomial_standard_deviation(double n, double p)</div><div class="ttdef"><b>Definition:</b> binomial_dist.cpp:36</div></div>
<div class="ttc" id="abinomial__dist_8cpp_html_a19ae0a6a2bd200fd1eb0e31b2bf4cc76"><div class="ttname"><a href="../../d6/db0/binomial__dist_8cpp.html#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> binomial_dist.cpp:65</div></div>
<div class="ttc" id="aupper_bound_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/algorithm/upper_bound.html">std::upper_bound</a></div><div class="ttdeci">T upper_bound(T... args)</div></div>
<div class="ttc" id="abinomial__dist_8cpp_html_a4416a7bc7fa87201883c54cdc4c82813"><div class="ttname"><a href="../../d6/db0/binomial__dist_8cpp.html#a4416a7bc7fa87201883c54cdc4c82813">binomial_expected</a></div><div class="ttdeci">double binomial_expected(double n, double p)</div><div class="ttdef"><b>Definition:</b> binomial_dist.cpp:22</div></div>
<div class="ttc" id="aendl_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a></div><div class="ttdeci">T endl(T... args)</div></div>
<div class="ttc" id="abinomial__dist_8cpp_html_a78d36635232e54b5d71fcbf1eac9a49a"><div class="ttname"><a href="../../d6/db0/binomial__dist_8cpp.html#a78d36635232e54b5d71fcbf1eac9a49a">nCr</a></div><div class="ttdeci">double nCr(double n, double r)</div><div class="ttdef"><b>Definition:</b> binomial_dist.cpp:47</div></div>
<div class="ttc" id="abinomial__dist_8cpp_html_acd4dd4558031e4c5d045c801f73d8861"><div class="ttname"><a href="../../d6/db0/binomial__dist_8cpp.html#acd4dd4558031e4c5d045c801f73d8861">binomial_variance</a></div><div class="ttdeci">double binomial_variance(double n, double p)</div><div class="ttdef"><b>Definition:</b> binomial_dist.cpp:29</div></div>
<div class="ttc" id="apow_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/math/pow.html">std::pow</a></div><div class="ttdeci">T pow(T... args)</div></div>
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