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| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.stat_tut.weg.geometric_eg"></a><a class="link" href="geometric_eg.html" title="Geometric Distribution Examples">Geometric Distribution |
| Examples</a> |
| </h4></div></div></div> |
| <p> |
| For this example, we will opt to #define two macros to control the error |
| and discrete handling policies. For this simple example, we want to avoid |
| throwing an exception (the default policy) and just return infinity. We |
| want to treat the distribution as if it was continuous, so we choose a |
| discrete_quantile policy of real, rather than the default policy integer_round_outwards. |
| </p> |
| <pre class="programlisting"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_OVERFLOW_ERROR_POLICY</span> <span class="identifier">ignore_error</span> |
| <span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_DISCRETE_QUANTILE_POLICY</span> <span class="identifier">real</span> |
| </pre> |
| <div class="caution"><table border="0" summary="Caution"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../doc/src/images/caution.png"></td> |
| <th align="left">Caution</th> |
| </tr> |
| <tr><td align="left" valign="top"><p> |
| It is vital to #include distributions etc <span class="bold"><strong>after</strong></span> |
| the above #defines |
| </p></td></tr> |
| </table></div> |
| <p> |
| After that we need some includes to provide easy access to the negative |
| binomial distribution, and we need some std library iostream, of course. |
| </p> |
| <pre class="programlisting"><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">geometric</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| <span class="comment">// for geometric_distribution</span> |
| <span class="keyword">using</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">geometric_distribution</span><span class="special">;</span> <span class="comment">//</span> |
| <span class="keyword">using</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">geometric</span><span class="special">;</span> <span class="comment">// typedef provides default type is double.</span> |
| <span class="keyword">using</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">pdf</span><span class="special">;</span> <span class="comment">// Probability mass function.</span> |
| <span class="keyword">using</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">cdf</span><span class="special">;</span> <span class="comment">// Cumulative density function.</span> |
| <span class="keyword">using</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">quantile</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">negative_binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| <span class="comment">// for negative_binomial_distribution</span> |
| <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">negative_binomial</span><span class="special">;</span> <span class="comment">// typedef provides default type is double.</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">normal</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| <span class="comment">// for negative_binomial_distribution</span> |
| <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">normal</span><span class="special">;</span> <span class="comment">// typedef provides default type is double.</span> |
| |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> |
| <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> |
| <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">noshowpoint</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">left</span><span class="special">;</span> |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iomanip</span><span class="special">></span> |
| <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">;</span> |
| |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">limits</span><span class="special">></span> |
| <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">;</span> |
| </pre> |
| <p> |
| It is always sensible to use try and catch blocks because defaults policies |
| are to throw an exception if anything goes wrong. |
| </p> |
| <p> |
| Simple try'n'catch blocks (see below) will ensure that you get a helpful |
| error message instead of an abrupt (and silent) program abort. |
| </p> |
| <h5> |
| <a name="math_toolkit.stat_tut.weg.geometric_eg.h0"></a> |
| <span class="phrase"><a name="math_toolkit.stat_tut.weg.geometric_eg.throwing_a_dice"></a></span><a class="link" href="geometric_eg.html#math_toolkit.stat_tut.weg.geometric_eg.throwing_a_dice">Throwing |
| a dice</a> |
| </h5> |
| <p> |
| The Geometric distribution describes the probability (<span class="emphasis"><em>p</em></span>) |
| of a number of failures to get the first success in <span class="emphasis"><em>k</em></span> |
| Bernoulli trials. (A <a href="http://en.wikipedia.org/wiki/Bernoulli_distribution" target="_top">Bernoulli |
| trial</a> is one with only two possible outcomes, success of failure, |
| and <span class="emphasis"><em>p</em></span> is the probability of success). |
| </p> |
| <p> |
| Suppose an 'fair' 6-face dice is thrown repeatedly: |
| </p> |
| <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">success_fraction</span> <span class="special">=</span> <span class="number">1.</span><span class="special">/</span><span class="number">6</span><span class="special">;</span> <span class="comment">// success_fraction (p) = 0.1666</span> |
| <span class="comment">// (so failure_fraction is 1 - success_fraction = 5./6 = 1- 0.1666 = 0.8333)</span> |
| </pre> |
| <p> |
| If the dice is thrown repeatedly until the <span class="bold"><strong>first</strong></span> |
| time a <span class="emphasis"><em>three</em></span> appears. The probablility distribution |
| of the number of times it is thrown <span class="bold"><strong>not</strong></span> |
| getting a <span class="emphasis"><em>three</em></span> (<span class="emphasis"><em>not-a-threes</em></span> |
| number of failures to get a <span class="emphasis"><em>three</em></span>) is a geometric |
| distribution with the success_fraction = 1/6 = 0.1666 ̇. |
| </p> |
| <p> |
| We therefore start by constructing a geometric distribution with the one |
| parameter success_fraction, the probability of success. |
| </p> |
| <pre class="programlisting"><span class="identifier">geometric</span> <span class="identifier">g6</span><span class="special">(</span><span class="identifier">success_fraction</span><span class="special">);</span> <span class="comment">// type double by default.</span> |
| </pre> |
| <p> |
| To confirm, we can echo the success_fraction parameter of the distribution. |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"success fraction of a six-sided dice is "</span> <span class="special"><<</span> <span class="identifier">g6</span><span class="special">.</span><span class="identifier">success_fraction</span><span class="special">()</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| So the probability of getting a three at the first throw (zero failures) |
| is |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.1667</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.1667</span> |
| </pre> |
| <p> |
| Note that the cdf and pdf are identical because the is only one throw. |
| If we want the probability of getting the first <span class="emphasis"><em>three</em></span> |
| on the 2nd throw: |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.1389</span> |
| </pre> |
| <p> |
| If we want the probability of getting the first <span class="emphasis"><em>three</em></span> |
| on the 1st or 2nd throw (allowing one failure): |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"pdf(g6, 0) + pdf(g6, 1) = "</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| Or more conveniently, and more generally, we can use the Cumulative Distribution |
| Function CDF. |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"cdf(g6, 1) = "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.3056</span> |
| </pre> |
| <p> |
| If we allow many more (12) throws, the probability of getting our <span class="emphasis"><em>three</em></span> |
| gets very high: |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"cdf(g6, 12) = "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">12</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.9065 or 90% probability.</span> |
| </pre> |
| <p> |
| If we want to be much more confident, say 99%, we can estimate the number |
| of throws to be this sure using the inverse or quantile. |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"quantile(g6, 0.99) = "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">0.99</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 24.26</span> |
| </pre> |
| <p> |
| Note that the value returned is not an integer: if you want an integer |
| result you should use either floor, round or ceil functions, or use the |
| policies mechanism. |
| </p> |
| <p> |
| See <a class="link" href="../../pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">understanding |
| discrete quantiles</a>. |
| </p> |
| <p> |
| The geometric distribution is related to the negative binomial    <code class="computeroutput"><span class="identifier">negative_binomial_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">RealType</span> |
| <span class="identifier">p</span><span class="special">);</span></code> |
| with parameter <span class="emphasis"><em>r</em></span> = 1. So we could get the same result |
| using the negative binomial, but using the geometric the results will be |
| faster, and may be more accurate. |
| </p> |
| <pre class="programlisting"><span class="identifier">negative_binomial</span> <span class="identifier">nb</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="identifier">success_fraction</span><span class="special">);</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.1389</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.3056</span> |
| </pre> |
| <p> |
| We could also the complement to express the required probability as 1 - |
| 0.99 = 0.01 (and get the same result): |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"quantile(complement(g6, 1 - p)) "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">0.01</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 24.26</span> |
| </pre> |
| <p> |
| Note too that Boost.Math geometric distribution is implemented as a continuous |
| function. Unlike other implementations (for example R) it <span class="bold"><strong>uses</strong></span> |
| the number of failures as a <span class="bold"><strong>real</strong></span> parameter, |
| not as an integer. If you want this integer behaviour, you may need to |
| enforce this by rounding the parameter you pass, probably rounding down, |
| to the nearest integer. For example, R returns the success fraction probability |
| for all values of failures from 0 to 0.999999 thus: |
| </p> |
| <pre class="programlisting">   R> formatC(pgeom(0.0001,0.5, FALSE), digits=17) " 0.5" |
| </pre> |
| <p> |
| So in Boost.Math the equivalent is |
| </p> |
| <pre class="programlisting"> <span class="identifier">geometric</span> <span class="identifier">g05</span><span class="special">(</span><span class="number">0.5</span><span class="special">);</span> <span class="comment">// Probability of success = 0.5 or 50%</span> |
| <span class="comment">// Output all potentially significant digits for the type, here double.</span> |
| |
| <span class="preprocessor">#ifdef</span> <span class="identifier">BOOST_NO_CXX11_NUMERIC_LIMITS</span> |
| <span class="keyword">int</span> <span class="identifier">max_digits10</span> <span class="special">=</span> <span class="number">2</span> <span class="special">+</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">policies</span><span class="special">::</span><span class="identifier">digits</span><span class="special"><</span><span class="keyword">double</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">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special"><></span> <span class="special">>()</span> <span class="special">*</span> <span class="number">30103UL</span><span class="special">)</span> <span class="special">/</span> <span class="number">100000UL</span><span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"BOOST_NO_CXX11_NUMERIC_LIMITS is defined"</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="preprocessor">#else</span> |
| <span class="keyword">int</span> <span class="identifier">max_digits10</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">;</span> |
| <span class="preprocessor">#endif</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Show all potentially significant decimal digits std::numeric_limits<double>::max_digits10 = "</span> |
| <span class="special"><<</span> <span class="identifier">max_digits10</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">max_digits10</span><span class="special">);</span> <span class="comment">//</span> |
| |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">g05</span><span class="special">,</span> <span class="number">0.0001</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// returns 0.5000346561579232, not exact 0.5.</span> |
| </pre> |
| <p> |
| To get the R discrete behaviour, you simply need to round with, for example, |
| the <code class="computeroutput"><span class="identifier">floor</span></code> function. |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">g05</span><span class="special">,</span> <span class="identifier">floor</span><span class="special">(</span><span class="number">0.0001</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// returns exactly 0.5</span> |
| </pre> |
| <pre class="programlisting"><code class="computeroutput"><span class="special">></span> <span class="identifier">formatC</span><span class="special">(</span><span class="identifier">pgeom</span><span class="special">(</span><span class="number">0.9999999</span><span class="special">,</span><span class="number">0.5</span><span class="special">,</span> <span class="identifier">FALSE</span><span class="special">),</span> <span class="identifier">digits</span><span class="special">=</span><span class="number">17</span><span class="special">)</span> <span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="string">" 0.25"</span></code> |
| <code class="computeroutput"><span class="special">></span> <span class="identifier">formatC</span><span class="special">(</span><span class="identifier">pgeom</span><span class="special">(</span><span class="number">1.999999</span><span class="special">,</span><span class="number">0.5</span><span class="special">,</span> <span class="identifier">FALSE</span><span class="special">),</span> <span class="identifier">digits</span><span class="special">=</span><span class="number">17</span><span class="special">)[</span><span class="number">1</span><span class="special">]</span> <span class="string">" 0.25"</span> <span class="identifier">k</span> <span class="special">=</span> <span class="number">1</span></code> |
| <code class="computeroutput"><span class="special">></span> <span class="identifier">formatC</span><span class="special">(</span><span class="identifier">pgeom</span><span class="special">(</span><span class="number">1.9999999</span><span class="special">,</span><span class="number">0.5</span><span class="special">,</span> <span class="identifier">FALSE</span><span class="special">),</span> <span class="identifier">digits</span><span class="special">=</span><span class="number">17</span><span class="special">)[</span><span class="number">1</span><span class="special">]</span> <span class="string">"0.12500000000000003"</span> <span class="identifier">k</span> <span class="special">=</span> <span class="number">2</span></code> |
| </pre> |
| <p> |
| shows that R makes an arbitrary round-up decision at about 1e7 from the |
| next integer above. This may be convenient in practice, and could be replicated |
| in C++ if desired. |
| </p> |
| <h5> |
| <a name="math_toolkit.stat_tut.weg.geometric_eg.h1"></a> |
| <span class="phrase"><a name="math_toolkit.stat_tut.weg.geometric_eg.surveying_customers_to_find_one_"></a></span><a class="link" href="geometric_eg.html#math_toolkit.stat_tut.weg.geometric_eg.surveying_customers_to_find_one_">Surveying |
| customers to find one with a faulty product</a> |
| </h5> |
| <p> |
| A company knows from warranty claims that 2% of their products will be |
| faulty, so the 'success_fraction' of finding a fault is 0.02. It wants |
| to interview a purchaser of faulty products to assess their 'user experience'. |
| </p> |
| <p> |
| To estimate how many customers they will probably need to contact in order |
| to find one who has suffered from the fault, we first construct a geometric |
| distribution with probability 0.02, and then chose a confidence, say 80%, |
| 95%, or 99% to finding a customer with a fault. Finally, we probably want |
| to round up the result to the integer above using the <code class="computeroutput"><span class="identifier">ceil</span></code> |
| function. (We could also use a policy, but that is hardly worthwhile for |
| this simple application.) |
| </p> |
| <p> |
| (This also assumes that each customer only buys one product: if customers |
| bought more than one item, the probability of finding a customer with a |
| fault obviously improves.) |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">5</span><span class="special">);</span> |
| <span class="identifier">geometric</span> <span class="identifier">g</span><span class="special">(</span><span class="number">0.02</span><span class="special">);</span> <span class="comment">// On average, 2 in 100 products are faulty.</span> |
| <span class="keyword">double</span> <span class="identifier">c</span> <span class="special">=</span> <span class="number">0.95</span><span class="special">;</span> <span class="comment">// 95% confidence.</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" quantile(g, "</span> <span class="special"><<</span> <span class="identifier">c</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="identifier">c</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"To be "</span> <span class="special"><<</span> <span class="identifier">c</span> <span class="special">*</span> <span class="number">100</span> |
| <span class="special"><<</span> <span class="string">"% confident of finding we customer with a fault, need to survey "</span> |
| <span class="special"><<</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="identifier">c</span><span class="special">))</span> <span class="special"><<</span> <span class="string">" customers."</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 148</span> |
| <span class="identifier">c</span> <span class="special">=</span> <span class="number">0.99</span><span class="special">;</span> <span class="comment">// Very confident.</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"To be "</span> <span class="special"><<</span> <span class="identifier">c</span> <span class="special">*</span> <span class="number">100</span> |
| <span class="special"><<</span> <span class="string">"% confident of finding we customer with a fault, need to survey "</span> |
| <span class="special"><<</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="identifier">c</span><span class="special">))</span> <span class="special"><<</span> <span class="string">" customers."</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 227</span> |
| <span class="identifier">c</span> <span class="special">=</span> <span class="number">0.80</span><span class="special">;</span> <span class="comment">// Only reasonably confident.</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"To be "</span> <span class="special"><<</span> <span class="identifier">c</span> <span class="special">*</span> <span class="number">100</span> |
| <span class="special"><<</span> <span class="string">"% confident of finding we customer with a fault, need to survey "</span> |
| <span class="special"><<</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="identifier">c</span><span class="special">))</span> <span class="special"><<</span> <span class="string">" customers."</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 79</span> |
| </pre> |
| <h5> |
| <a name="math_toolkit.stat_tut.weg.geometric_eg.h2"></a> |
| <span class="phrase"><a name="math_toolkit.stat_tut.weg.geometric_eg.basket_ball_shooters"></a></span><a class="link" href="geometric_eg.html#math_toolkit.stat_tut.weg.geometric_eg.basket_ball_shooters">Basket |
| Ball Shooters</a> |
| </h5> |
| <p> |
| According to Wikipedia, average pro basket ball players get <a href="http://en.wikipedia.org/wiki/Free_throw" target="_top">free |
| throws</a> in the baskets 70 to 80 % of the time, but some get as high |
| as 95%, and others as low as 50%. Suppose we want to compare the probabilities |
| of failing to get a score only on the first or on the fifth shot? To start |
| we will consider the average shooter, say 75%. So we construct a geometric |
| distribution with success_fraction parameter 75/100 = 0.75. |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">2</span><span class="special">);</span> |
| <span class="identifier">geometric</span> <span class="identifier">gav</span><span class="special">(</span><span class="number">0.75</span><span class="special">);</span> <span class="comment">// Shooter averages 7.5 out of 10 in the basket.</span> |
| </pre> |
| <p> |
| What is probability of getting 1st try in the basket, that is with no failures? |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of score on 1st try = "</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">gav</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.75</span> |
| </pre> |
| <p> |
| This is, of course, the success_fraction probability 75%. What is the probability |
| that the shooter only scores on the fifth shot? So there are 5-1 = 4 failures |
| before the first success. |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of score on 5th try = "</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">gav</span><span class="special">,</span> <span class="number">4</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.0029</span> |
| </pre> |
| <p> |
| Now compare this with the poor and the best players success fraction. We |
| need to constructing new distributions with the different success fractions, |
| and then get the corresponding probability density functions values: |
| </p> |
| <pre class="programlisting"><span class="identifier">geometric</span> <span class="identifier">gbest</span><span class="special">(</span><span class="number">0.95</span><span class="special">);</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of score on 5th try = "</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">gbest</span><span class="special">,</span> <span class="number">4</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 5.9e-6</span> |
| <span class="identifier">geometric</span> <span class="identifier">gmediocre</span><span class="special">(</span><span class="number">0.50</span><span class="special">);</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of score on 5th try = "</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">gmediocre</span><span class="special">,</span> <span class="number">4</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.031</span> |
| </pre> |
| <p> |
| So we can see the very much smaller chance (0.000006) of 4 failures by |
| the best shooters, compared to the 0.03 of the mediocre. |
| </p> |
| <h5> |
| <a name="math_toolkit.stat_tut.weg.geometric_eg.h3"></a> |
| <span class="phrase"><a name="math_toolkit.stat_tut.weg.geometric_eg.estimating_failures"></a></span><a class="link" href="geometric_eg.html#math_toolkit.stat_tut.weg.geometric_eg.estimating_failures">Estimating |
| failures</a> |
| </h5> |
| <p> |
| Of course one man's failure is an other man's success. So a fault can be |
| defined as a 'success'. |
| </p> |
| <p> |
| If a fault occurs once after 100 flights, then one might naively say that |
| the risk of fault is obviously 1 in 100 = 1/100, a probability of 0.01. |
| </p> |
| <p> |
| This is the best estimate we can make, but while it is the truth, it is |
| not the whole truth, for it hides the big uncertainty when estimating from |
| a single event. "One swallow doesn't make a summer." To show |
| the magnitude of the uncertainty, the geometric (or the negative binomial) |
| distribution can be used. |
| </p> |
| <p> |
| If we chose the popular 95% confidence in the limits, corresponding to |
| an alpha of 0.05, because we are calculating a two-sided interval, we must |
| divide alpha by two. |
| </p> |
| <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha</span> <span class="special">=</span> <span class="number">0.05</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">k</span> <span class="special">=</span> <span class="number">100</span><span class="special">;</span> <span class="comment">// So frequency of occurence is 1/100.</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability is failure is "</span> <span class="special"><<</span> <span class="number">1</span><span class="special">/</span><span class="identifier">k</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">geometric</span><span class="special">::</span><span class="identifier">find_lower_bound_on_p</span><span class="special">(</span><span class="identifier">k</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span><span class="special">);</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"geometric::find_lower_bound_on_p("</span> <span class="special"><<</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">k</span><span class="special">)</span> <span class="special"><<</span> <span class="string">", "</span> <span class="special"><<</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span> <span class="special"><<</span> <span class="string">") = "</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="comment">// 0.00025</span> |
| <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">geometric</span><span class="special">::</span><span class="identifier">find_upper_bound_on_p</span><span class="special">(</span><span class="identifier">k</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span><span class="special">);</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"geometric::find_upper_bound_on_p("</span> <span class="special"><<</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">k</span><span class="special">)</span> <span class="special"><<</span> <span class="string">", "</span> <span class="special"><<</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span> <span class="special"><<</span> <span class="string">") = "</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="comment">// 0.037</span> |
| </pre> |
| <p> |
| So while we estimate the probability is 0.01, it might lie between 0.0003 |
| and 0.04. Even if we relax our confidence to alpha = 90%, the bounds only |
| contract to 0.0005 and 0.03. And if we require a high confidence, they |
| widen to 0.00005 to 0.05. |
| </p> |
| <pre class="programlisting"><span class="identifier">alpha</span> <span class="special">=</span> <span class="number">0.1</span><span class="special">;</span> <span class="comment">// 90% confidence.</span> |
| <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">geometric</span><span class="special">::</span><span class="identifier">find_lower_bound_on_p</span><span class="special">(</span><span class="identifier">k</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span><span class="special">);</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"geometric::find_lower_bound_on_p("</span> <span class="special"><<</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">k</span><span class="special">)</span> <span class="special"><<</span> <span class="string">", "</span> <span class="special"><<</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span> <span class="special"><<</span> <span class="string">") = "</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="comment">// 0.0005</span> |
| <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">geometric</span><span class="special">::</span><span class="identifier">find_upper_bound_on_p</span><span class="special">(</span><span class="identifier">k</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span><span class="special">);</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"geometric::find_upper_bound_on_p("</span> <span class="special"><<</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">k</span><span class="special">)</span> <span class="special"><<</span> <span class="string">", "</span> <span class="special"><<</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span> <span class="special"><<</span> <span class="string">") = "</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="comment">// 0.03</span> |
| |
| <span class="identifier">alpha</span> <span class="special">=</span> <span class="number">0.01</span><span class="special">;</span> <span class="comment">// 99% confidence.</span> |
| <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">geometric</span><span class="special">::</span><span class="identifier">find_lower_bound_on_p</span><span class="special">(</span><span class="identifier">k</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span><span class="special">);</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"geometric::find_lower_bound_on_p("</span> <span class="special"><<</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">k</span><span class="special">)</span> <span class="special"><<</span> <span class="string">", "</span> <span class="special"><<</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span> <span class="special"><<</span> <span class="string">") = "</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="comment">// 5e-005</span> |
| <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">geometric</span><span class="special">::</span><span class="identifier">find_upper_bound_on_p</span><span class="special">(</span><span class="identifier">k</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span><span class="special">);</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"geometric::find_upper_bound_on_p("</span> <span class="special"><<</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">k</span><span class="special">)</span> <span class="special"><<</span> <span class="string">", "</span> <span class="special"><<</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span> <span class="special"><<</span> <span class="string">") = "</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="comment">// 0.052</span> |
| </pre> |
| <p> |
| In real life, there will usually be more than one event (fault or success), |
| when the negative binomial, which has the neccessary extra parameter, will |
| be needed. |
| </p> |
| <p> |
| As noted above, using a catch block is always a good idea, even if you |
| hope not to use it! |
| </p> |
| <pre class="programlisting"><span class="special">}</span> |
| <span class="keyword">catch</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span><span class="special">&</span> <span class="identifier">e</span><span class="special">)</span> |
| <span class="special">{</span> <span class="comment">// Since we have set an overflow policy of ignore_error,</span> |
| <span class="comment">// an overflow exception should never be thrown.</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\nMessage from thrown exception was:\n "</span> <span class="special"><<</span> <span class="identifier">e</span><span class="special">.</span><span class="identifier">what</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> |
| </pre> |
| <p> |
| For example, without a ignore domain error policy, if we asked for |
| </p> |
| <pre class="programlisting"><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="special">-</span><span class="number">1</span><span class="special">)</span></pre> |
| <p> |
| for example, we would get an unhelpful abort, but with a catch: |
| </p> |
| <pre class="programlisting">Message from thrown exception was: |
| Error in function boost::math::pdf(const exponential_distribution<double>&, double): |
| Number of failures argument is -1, but must be >= 0 ! |
| </pre> |
| <p> |
| See full source C++ of this example at <a href="../../../../../example/geometric_examples.cpp" target="_top">geometric_examples.cpp</a> |
| </p> |
| <p> |
| <a class="link" href="neg_binom_eg/neg_binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for the Negative Binomial Distribution">See |
| negative_binomial confidence interval example.</a> |
| </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 © 2006-2010, 2012-2014 Nikhar Agrawal, |
| Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert |
| Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, |
| Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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> |
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