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| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h5 class="title"> |
| <a name="math_toolkit.dist.dist_ref.dists.geometric_dist"></a><a class="link" href="geometric_dist.html" title="Geometric Distribution">Geometric |
| Distribution</a> |
| </h5></div></div></div> |
| <p> |
| |
| </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></pre> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span> |
| <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a> <span class="special">=</span> <a class="link" href="../../../policy/pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span> |
| <span class="keyword">class</span> <span class="identifier">geometric_distribution</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">geometric_distribution</span><span class="special"><></span> <span class="identifier">geometric</span><span class="special">;</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <span class="keyword">class</span> <span class="identifier">geometric_distribution</span> |
| <span class="special">{</span> |
| <span class="keyword">public</span><span class="special">:</span> |
| <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> |
| <span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span> |
| <span class="comment">// Constructor from success_fraction: |
| </span> <span class="identifier">geometric_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">);</span> |
| |
| <span class="comment">// Parameter accessors: |
| </span> <span class="identifier">RealType</span> <span class="identifier">success_fraction</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> |
| <span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> |
| |
| <span class="comment">// Bounds on success fraction: |
| </span> <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_lower_bound_on_p</span><span class="special">(</span> |
| <span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">,</span> |
| <span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">,</span> |
| <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// alpha |
| </span> <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_upper_bound_on_p</span><span class="special">(</span> |
| <span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">,</span> |
| <span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">,</span> |
| <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// alpha |
| </span> |
| <span class="comment">// Estimate min/max number of trials: |
| </span> <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_minimum_number_of_trials</span><span class="special">(</span> |
| <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span> <span class="comment">// Number of failures. |
| </span> <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// Success fraction. |
| </span> <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// Probability threshold alpha. |
| </span> <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_maximum_number_of_trials</span><span class="special">(</span> |
| <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span> <span class="comment">// Number of failures. |
| </span> <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// Success fraction. |
| </span> <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// Probability threshold alpha. |
| </span><span class="special">};</span> |
| |
| <span class="special">}}</span> <span class="comment">// namespaces |
| </span></pre> |
| <p> |
| The class type <code class="computeroutput"><span class="identifier">geometric_distribution</span></code> |
| represents a <a href="http://en.wikipedia.org/wiki/geometric_distribution" target="_top">geometric |
| distribution</a>: it is used when there are exactly two mutually |
| exclusive outcomes of a <a href="http://en.wikipedia.org/wiki/Bernoulli_trial" target="_top">Bernoulli |
| trial</a>: these outcomes are labelled "success" and "failure". |
| </p> |
| <p> |
| For <a href="http://en.wikipedia.org/wiki/Bernoulli_trial" target="_top">Bernoulli |
| trials</a> each with success fraction <span class="emphasis"><em>p</em></span>, the |
| geometric distribution gives the probability of observing <span class="emphasis"><em>k</em></span> |
| trials (failures, events, occurrences, or arrivals) before the first |
| success. |
| </p> |
| <div class="note"><table border="0" summary="Note"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../../doc/src/images/note.png"></td> |
| <th align="left">Note</th> |
| </tr> |
| <tr><td align="left" valign="top"><p> |
| For this implementation, the set of trials <span class="bold"><strong>includes |
| zero</strong></span> (unlike another definition where the set of trials |
| starts at one, sometimes named <span class="emphasis"><em>shifted</em></span>). |
| </p></td></tr> |
| </table></div> |
| <p> |
| The geometric distribution assumes that success_fraction <span class="emphasis"><em>p</em></span> |
| is fixed for all <span class="emphasis"><em>k</em></span> trials. |
| </p> |
| <p> |
| The probability that there are <span class="emphasis"><em>k</em></span> failures before |
| the first success is |
| </p> |
| <p> |
|    Pr(Y=<span class="emphasis"><em>k</em></span>) = (1-<span class="emphasis"><em>p</em></span>)<sup><span class="emphasis"><em>k</em></span></sup><span class="emphasis"><em>p</em></span> |
| </p> |
| <p> |
| For example, when throwing a 6-face dice the success probability <span class="emphasis"><em>p</em></span> |
| = 1/6 = 0.1666 ̇  . Throwing repeatedly until a <span class="emphasis"><em>three</em></span> |
| appears, the probability distribution of the number of times <span class="emphasis"><em>not-a-three</em></span> |
| is thrown is geometric. |
| </p> |
| <p> |
| Geometric distribution has the Probability Density Function PDF: |
| </p> |
| <p> |
|    (1-<span class="emphasis"><em>p</em></span>)<sup><span class="emphasis"><em>k</em></span></sup><span class="emphasis"><em>p</em></span> |
| </p> |
| <p> |
| The following graph illustrates how the PDF and CDF vary for three examples |
| of the success fraction <span class="emphasis"><em>p</em></span>, (when considering the |
| geometric distribution as a continuous function), |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../graphs/geometric_pdf_2.png" align="middle"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../graphs/geometric_cdf_2.png" align="middle"></span> |
| </p> |
| <p> |
| and as discrete. |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../graphs/geometric_pdf_discrete.png" align="middle"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../graphs/geometric_cdf_discrete.png" align="middle"></span> |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.geometric_dist.related_distributions"></a><h5> |
| <a name="id1181954"></a> |
| <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_dist.related_distributions">Related |
| Distributions</a> |
| </h5> |
| <p> |
| The geometric distribution is a special case of the <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative |
| Binomial Distribution</a> with successes parameter <span class="emphasis"><em>r</em></span> |
| = 1, so only one first and only success is required : thus by definition |
|    <code class="computeroutput"><span class="identifier">geometric</span><span class="special">(</span><span class="identifier">p</span><span class="special">)</span> <span class="special">==</span> <span class="identifier">negative_binomial</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> |
| <span class="identifier">p</span><span class="special">)</span></code> |
| </p> |
| <pre class="programlisting"><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">success_fraction</span><span class="special">);</span> |
| <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">geometric</span> <span class="identifier">g</span><span class="special">(</span><span class="identifier">success_fraction</span><span class="special">);</span> |
| <span class="identifier">ASSERT</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">pdf</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="number">1</span><span class="special">));</span> |
| </pre> |
| <p> |
| This implementation uses real numbers for the computation throughout |
| (because it uses the <span class="bold"><strong>real-valued</strong></span> power |
| and exponential functions). So to obtain a conventional strictly-discrete |
| geometric distribution you must ensure that an integer value is provided |
| for the number of trials (random variable) <span class="emphasis"><em>k</em></span>, and |
| take integer values (floor or ceil functions) from functions that return |
| a number of successes. |
| </p> |
| <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> |
| The geometric distribution is a discrete distribution: internally, |
| functions like the <code class="computeroutput"><span class="identifier">cdf</span></code> |
| and <code class="computeroutput"><span class="identifier">pdf</span></code> are treated |
| "as if" they are continuous functions, but in reality the |
| results returned from these functions only have meaning if an integer |
| value is provided for the random variate argument. |
| </p> |
| <p> |
| The quantile function will by default return an integer result that |
| has been <span class="emphasis"><em>rounded outwards</em></span>. That is to say lower |
| quantiles (where the probability is less than 0.5) are rounded downward, |
| and upper quantiles (where the probability is greater than 0.5) are |
| rounded upwards. This behaviour ensures that if an X% quantile is requested, |
| then <span class="emphasis"><em>at least</em></span> the requested coverage will be present |
| in the central region, and <span class="emphasis"><em>no more than</em></span> the requested |
| coverage will be present in the tails. |
| </p> |
| <p> |
| This behaviour can be changed so that the quantile functions are rounded |
| differently, or even return a real-valued result using <a class="link" href="../../../policy/pol_overview.html" title="Policy Overview">Policies</a>. |
| It is strongly recommended that you read the tutorial <a class="link" href="../../../policy/pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">Understanding |
| Quantiles of Discrete Distributions</a> before using the quantile |
| function on the geometric distribution. The <a class="link" href="../../../policy/pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">reference |
| docs</a> describe how to change the rounding policy for these distributions. |
| </p> |
| </td></tr> |
| </table></div> |
| <a name="math_toolkit.dist.dist_ref.dists.geometric_dist.member_functions"></a><h5> |
| <a name="id1182240"></a> |
| <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_dist.member_functions">Member |
| Functions</a> |
| </h5> |
| <a name="math_toolkit.dist.dist_ref.dists.geometric_dist.constructor"></a><h6> |
| <a name="id1182256"></a> |
| <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_dist.constructor">Constructor</a> |
| </h6> |
| <pre class="programlisting"><span class="identifier">geometric_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">);</span> |
| </pre> |
| <p> |
| Constructor: <span class="emphasis"><em>p</em></span> or success_fraction is the probability |
| of success of a single trial. |
| </p> |
| <p> |
| Requires: <code class="computeroutput"><span class="number">0</span> <span class="special"><=</span> |
| <span class="identifier">p</span> <span class="special"><=</span> |
| <span class="number">1</span></code>. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.geometric_dist.accessors"></a><h6> |
| <a name="id1182336"></a> |
| <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_dist.accessors">Accessors</a> |
| </h6> |
| <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">success_fraction</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> <span class="comment">// successes / trials (0 <= p <= 1) |
| </span></pre> |
| <p> |
| Returns the success_fraction parameter <span class="emphasis"><em>p</em></span> from which |
| this distribution was constructed. |
| </p> |
| <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">successes</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> <span class="comment">// required successes always one, |
| </span><span class="comment">// included for compatibility with negative binomial distribution |
| </span><span class="comment">// with successes r == 1. |
| </span></pre> |
| <p> |
| Returns unity. |
| </p> |
| <p> |
| The following functions are equivalent to those provided for the negative |
| binomial, with successes = 1, but are provided here for completeness. |
| </p> |
| <p> |
| The best method of calculation for the following functions is disputed: |
| see <a class="link" href="binomial_dist.html" title="Binomial Distribution">Binomial |
| Distribution</a> and <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative |
| Binomial Distribution</a> for more discussion. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.geometric_dist.lower_bound_on_success_fraction_parameter__emphasis_p__emphasis_"></a><h6> |
| <a name="id1182444"></a> |
| <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_dist.lower_bound_on_success_fraction_parameter__emphasis_p__emphasis_">Lower |
| Bound on success_fraction Parameter <span class="emphasis"><em>p</em></span></a> |
| </h6> |
| <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_lower_bound_on_p</span><span class="special">(</span> |
| <span class="identifier">RealType</span> <span class="identifier">failures</span><span class="special">,</span> |
| <span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">)</span> <span class="comment">// (0 <= alpha <= 1), 0.05 equivalent to 95% confidence. |
| </span></pre> |
| <p> |
| Returns a <span class="bold"><strong>lower bound</strong></span> on the success |
| fraction: |
| </p> |
| <div class="variablelist"> |
| <p class="title"><b></b></p> |
| <dl> |
| <dt><span class="term">failures</span></dt> |
| <dd><p> |
| The total number of failures before the 1st success. |
| </p></dd> |
| <dt><span class="term">alpha</span></dt> |
| <dd><p> |
| The largest acceptable probability that the true value of the success |
| fraction is <span class="bold"><strong>less than</strong></span> the value |
| returned. |
| </p></dd> |
| </dl> |
| </div> |
| <p> |
| For example, if you observe <span class="emphasis"><em>k</em></span> failures from <span class="emphasis"><em>n</em></span> |
| trials the best estimate for the success fraction is simply 1/<span class="emphasis"><em>n</em></span>, |
| but if you want to be 95% sure that the true value is <span class="bold"><strong>greater |
| than</strong></span> some value, <span class="emphasis"><em>p<sub>min</sub></em></span>, then: |
| </p> |
| <pre class="programlisting"><span class="identifier">p</span><sub>min</sub> <span class="special">=</span> <span class="identifier">geometric_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span> |
| <span class="identifier">find_lower_bound_on_p</span><span class="special">(</span><span class="identifier">failures</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span> |
| </pre> |
| <p> |
| <a class="link" href="../../stat_tut/weg/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> |
| <p> |
| This function uses the Clopper-Pearson method of computing the lower |
| bound on the success fraction, whilst many texts refer to this method |
| as giving an "exact" result in practice it produces an interval |
| that guarantees <span class="emphasis"><em>at least</em></span> the coverage required, |
| and may produce pessimistic estimates for some combinations of <span class="emphasis"><em>failures</em></span> |
| and <span class="emphasis"><em>successes</em></span>. See: |
| </p> |
| <p> |
| <a href="http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf" target="_top">Yong |
| Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some |
| Discrete Distributions. Computational statistics and data analysis, 2005, |
| vol. 48, no3, 605-621</a>. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.geometric_dist.upper_bound_on_success_fraction_parameter_p"></a><h6> |
| <a name="id1182673"></a> |
| <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_dist.upper_bound_on_success_fraction_parameter_p">Upper |
| Bound on success_fraction Parameter p</a> |
| </h6> |
| <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_upper_bound_on_p</span><span class="special">(</span> |
| <span class="identifier">RealType</span> <span class="identifier">trials</span><span class="special">,</span> |
| <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// (0 <= alpha <= 1), 0.05 equivalent to 95% confidence. |
| </span></pre> |
| <p> |
| Returns an <span class="bold"><strong>upper bound</strong></span> on the success |
| fraction: |
| </p> |
| <div class="variablelist"> |
| <p class="title"><b></b></p> |
| <dl> |
| <dt><span class="term">trials</span></dt> |
| <dd><p> |
| The total number of trials conducted. |
| </p></dd> |
| <dt><span class="term">alpha</span></dt> |
| <dd><p> |
| The largest acceptable probability that the true value of the success |
| fraction is <span class="bold"><strong>greater than</strong></span> the value |
| returned. |
| </p></dd> |
| </dl> |
| </div> |
| <p> |
| For example, if you observe <span class="emphasis"><em>k</em></span> successes from <span class="emphasis"><em>n</em></span> |
| trials the best estimate for the success fraction is simply <span class="emphasis"><em>k/n</em></span>, |
| but if you want to be 95% sure that the true value is <span class="bold"><strong>less |
| than</strong></span> some value, <span class="emphasis"><em>p<sub>max</sub></em></span>, then: |
| </p> |
| <pre class="programlisting"><span class="identifier">p</span><sub>max</sub> <span class="special">=</span> <span class="identifier">geometric_distribution</span><span class="special"><</span><span class="identifier">RealType</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="number">0.05</span><span class="special">);</span> |
| </pre> |
| <p> |
| <a class="link" href="../../stat_tut/weg/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> |
| <p> |
| This function uses the Clopper-Pearson method of computing the lower |
| bound on the success fraction, whilst many texts refer to this method |
| as giving an "exact" result in practice it produces an interval |
| that guarantees <span class="emphasis"><em>at least</em></span> the coverage required, |
| and may produce pessimistic estimates for some combinations of <span class="emphasis"><em>failures</em></span> |
| and <span class="emphasis"><em>successes</em></span>. See: |
| </p> |
| <p> |
| <a href="http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf" target="_top">Yong |
| Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some |
| Discrete Distributions. Computational statistics and data analysis, 2005, |
| vol. 48, no3, 605-621</a>. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.geometric_dist.estimating_number_of_trials_to_ensure_at_least_a_certain_number_of_failures"></a><h6> |
| <a name="id1182900"></a> |
| <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_dist.estimating_number_of_trials_to_ensure_at_least_a_certain_number_of_failures">Estimating |
| Number of Trials to Ensure at Least a Certain Number of Failures</a> |
| </h6> |
| <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_minimum_number_of_trials</span><span class="special">(</span> |
| <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span> <span class="comment">// number of failures. |
| </span> <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// success fraction. |
| </span> <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// probability threshold (0.05 equivalent to 95%). |
| </span></pre> |
| <p> |
| This functions estimates the number of trials required to achieve a certain |
| probability that <span class="bold"><strong>more than <span class="emphasis"><em>k</em></span> |
| failures will be observed</strong></span>. |
| </p> |
| <div class="variablelist"> |
| <p class="title"><b></b></p> |
| <dl> |
| <dt><span class="term">k</span></dt> |
| <dd><p> |
| The target number of failures to be observed. |
| </p></dd> |
| <dt><span class="term">p</span></dt> |
| <dd><p> |
| The probability of <span class="emphasis"><em>success</em></span> for each trial. |
| </p></dd> |
| <dt><span class="term">alpha</span></dt> |
| <dd><p> |
| The maximum acceptable <span class="emphasis"><em>risk</em></span> that only <span class="emphasis"><em>k</em></span> |
| failures or fewer will be observed. |
| </p></dd> |
| </dl> |
| </div> |
| <p> |
| For example: |
| </p> |
| <pre class="programlisting"><span class="identifier">geometric_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span><span class="identifier">find_minimum_number_of_trials</span><span class="special">(</span><span class="number">10</span><span class="special">,</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the smallest number of trials we must conduct to be 95% (1-0.05) |
| sure of seeing 10 failures that occur with frequency one half. |
| </p> |
| <p> |
| <a class="link" href="../../stat_tut/weg/neg_binom_eg/neg_binom_size_eg.html" title="Estimating Sample Sizes for the Negative Binomial.">Worked |
| Example.</a> |
| </p> |
| <p> |
| This function uses numeric inversion of the geometric distribution to |
| obtain the result: another interpretation of the result is that it finds |
| the number of trials (failures) that will lead to an <span class="emphasis"><em>alpha</em></span> |
| probability of observing <span class="emphasis"><em>k</em></span> failures or fewer. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.geometric_dist.estimating_number_of_trials_to_ensure_a_maximum_number_of_failures_or_less"></a><h6> |
| <a name="id1183140"></a> |
| <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_dist.estimating_number_of_trials_to_ensure_a_maximum_number_of_failures_or_less">Estimating |
| Number of Trials to Ensure a Maximum Number of Failures or Less</a> |
| </h6> |
| <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_maximum_number_of_trials</span><span class="special">(</span> |
| <span class="identifier">RealType</span> <span class="identifier">k</span><span class="special">,</span> <span class="comment">// number of failures. |
| </span> <span class="identifier">RealType</span> <span class="identifier">p</span><span class="special">,</span> <span class="comment">// success fraction. |
| </span> <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">);</span> <span class="comment">// probability threshold (0.05 equivalent to 95%). |
| </span></pre> |
| <p> |
| This functions estimates the maximum number of trials we can conduct |
| and achieve a certain probability that <span class="bold"><strong>k failures |
| or fewer will be observed</strong></span>. |
| </p> |
| <div class="variablelist"> |
| <p class="title"><b></b></p> |
| <dl> |
| <dt><span class="term">k</span></dt> |
| <dd><p> |
| The maximum number of failures to be observed. |
| </p></dd> |
| <dt><span class="term">p</span></dt> |
| <dd><p> |
| The probability of <span class="emphasis"><em>success</em></span> for each trial. |
| </p></dd> |
| <dt><span class="term">alpha</span></dt> |
| <dd><p> |
| The maximum acceptable <span class="emphasis"><em>risk</em></span> that more than |
| <span class="emphasis"><em>k</em></span> failures will be observed. |
| </p></dd> |
| </dl> |
| </div> |
| <p> |
| For example: |
| </p> |
| <pre class="programlisting"><span class="identifier">geometric_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span><span class="identifier">find_maximum_number_of_trials</span><span class="special">(</span><span class="number">0</span><span class="special">,</span> <span class="number">1.0</span><span class="special">-</span><span class="number">1.0</span><span class="special">/</span><span class="number">1000000</span><span class="special">,</span> <span class="number">0.05</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the largest number of trials we can conduct and still be 95% |
| sure of seeing no failures that occur with frequency one in one million. |
| </p> |
| <p> |
| This function uses numeric inversion of the geometric distribution to |
| obtain the result: another interpretation of the result, is that it finds |
| the number of trials that will lead to an <span class="emphasis"><em>alpha</em></span> |
| probability of observing more than k failures. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.geometric_dist.non_member_accessors"></a><h5> |
| <a name="id1184279"></a> |
| <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_dist.non_member_accessors">Non-member |
| Accessors</a> |
| </h5> |
| <p> |
| All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member |
| accessor functions</a> that are generic to all distributions are supported: |
| <a class="link" href="../nmp.html#math.dist.cdf">Cumulative Distribution Function</a>, |
| <a class="link" href="../nmp.html#math.dist.pdf">Probability Density Function</a>, <a class="link" href="../nmp.html#math.dist.quantile">Quantile</a>, <a class="link" href="../nmp.html#math.dist.hazard">Hazard |
| Function</a>, <a class="link" href="../nmp.html#math.dist.chf">Cumulative Hazard Function</a>, |
| <a class="link" href="../nmp.html#math.dist.mean">mean</a>, <a class="link" href="../nmp.html#math.dist.median">median</a>, |
| <a class="link" href="../nmp.html#math.dist.mode">mode</a>, <a class="link" href="../nmp.html#math.dist.variance">variance</a>, |
| <a class="link" href="../nmp.html#math.dist.sd">standard deviation</a>, <a class="link" href="../nmp.html#math.dist.skewness">skewness</a>, |
| <a class="link" href="../nmp.html#math.dist.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a>, |
| <a class="link" href="../nmp.html#math.dist.range">range</a> and <a class="link" href="../nmp.html#math.dist.support">support</a>. |
| </p> |
| <p> |
| However it's worth taking a moment to define what these actually mean |
| in the context of this distribution: |
| </p> |
| <div class="table"> |
| <a name="math_toolkit.dist.dist_ref.dists.geometric_dist.meaning_of_the_non_member_accessors_"></a><p class="title"><b>Table 14. Meaning of the non-member accessors.</b></p> |
| <div class="table-contents"><table class="table" summary="Meaning of the non-member accessors."> |
| <colgroup> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Function |
| </p> |
| </th> |
| <th> |
| <p> |
| Meaning |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| <a class="link" href="../nmp.html#math.dist.pdf">Probability Density Function</a> |
| </p> |
| </td> |
| <td> |
| <p> |
| The probability of obtaining <span class="bold"><strong>exactly |
| k failures</strong></span> from <span class="emphasis"><em>k</em></span> trials with |
| success fraction p. For example: |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">geometric</span><span class="special">(</span><span class="identifier">p</span><span class="special">),</span> <span class="identifier">k</span><span class="special">)</span></pre> |
| <p> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| <a class="link" href="../nmp.html#math.dist.cdf">Cumulative Distribution Function</a> |
| </p> |
| </td> |
| <td> |
| <p> |
| The probability of obtaining <span class="bold"><strong>k failures |
| or fewer</strong></span> from <span class="emphasis"><em>k</em></span> trials with |
| success fraction p and success on the last trial. For example: |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">geometric</span><span class="special">(</span><span class="identifier">p</span><span class="special">),</span> <span class="identifier">k</span><span class="special">)</span></pre> |
| <p> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| <a class="link" href="../nmp.html#math.dist.ccdf">Complement of the Cumulative |
| Distribution Function</a> |
| </p> |
| </td> |
| <td> |
| <p> |
| The probability of obtaining <span class="bold"><strong>more than |
| k failures</strong></span> from <span class="emphasis"><em>k</em></span> trials with |
| success fraction p and success on the last trial. For example: |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">geometric</span><span class="special">(</span><span class="identifier">p</span><span class="special">),</span> <span class="identifier">k</span><span class="special">))</span></pre> |
| <p> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| <a class="link" href="../nmp.html#math.dist.quantile">Quantile</a> |
| </p> |
| </td> |
| <td> |
| <p> |
| The <span class="bold"><strong>greatest</strong></span> number of failures |
| <span class="emphasis"><em>k</em></span> expected to be observed from <span class="emphasis"><em>k</em></span> |
| trials with success fraction <span class="emphasis"><em>p</em></span>, at probability |
| <span class="emphasis"><em>P</em></span>. Note that the value returned is a real-number, |
| and not an integer. Depending on the use case you may want |
| to take either the floor or ceiling of the real result. For |
| example: |
| </p> |
| <pre class="programlisting"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">geometric</span><span class="special">(</span><span class="identifier">p</span><span class="special">),</span> <span class="identifier">P</span><span class="special">)</span></pre> |
| <p> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| <a class="link" href="../nmp.html#math.dist.quantile_c">Quantile from the complement |
| of the probability</a> |
| </p> |
| </td> |
| <td> |
| <p> |
| The <span class="bold"><strong>smallest</strong></span> number of failures |
| <span class="emphasis"><em>k</em></span> expected to be observed from <span class="emphasis"><em>k</em></span> |
| trials with success fraction <span class="emphasis"><em>p</em></span>, at probability |
| <span class="emphasis"><em>P</em></span>. Note that the value returned is a real-number, |
| and not an integer. Depending on the use case you may want |
| to take either the floor or ceiling of the real result. For |
| example: |
| </p> |
| <pre class="programlisting"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">geometric</span><span class="special">(</span><span class="identifier">p</span><span class="special">),</span> <span class="identifier">P</span><span class="special">))</span></pre> |
| <p> |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><a name="math_toolkit.dist.dist_ref.dists.geometric_dist.accuracy"></a><h5> |
| <a name="id1184789"></a> |
| <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_dist.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| This distribution is implemented using the pow and exp functions, so |
| most results are accurate within a few epsilon for the RealType. For |
| extreme values of <code class="computeroutput"><span class="keyword">double</span></code> |
| <span class="emphasis"><em>p</em></span>, for example 0.9999999999, accuracy can fall significantly, |
| for example to 10 decimal digits (from 16). |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.geometric_dist.implementation"></a><h5> |
| <a name="id1184822"></a> |
| <a class="link" href="geometric_dist.html#math_toolkit.dist.dist_ref.dists.geometric_dist.implementation">Implementation</a> |
| </h5> |
| <p> |
| In the following table, <span class="emphasis"><em>p</em></span> is the probability that |
| any one trial will be successful (the success fraction), <span class="emphasis"><em>k</em></span> |
| is the number of failures, <span class="emphasis"><em>p</em></span> is the probability |
| and <span class="emphasis"><em>q = 1-p</em></span>, <span class="emphasis"><em>x</em></span> is the given |
| probability to estimate the expected number of failures using the quantile. |
| </p> |
| <div class="informaltable"><table class="table"> |
| <colgroup> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Function |
| </p> |
| </th> |
| <th> |
| <p> |
| Implementation Notes |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| pdf |
| </p> |
| </td> |
| <td> |
| <p> |
| pdf = p * pow(q, k) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf |
| </p> |
| </td> |
| <td> |
| <p> |
| cdf = 1 - q<sup>k=1</sup> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf complement |
| </p> |
| </td> |
| <td> |
| <p> |
| exp(log1p(-p) * (k+1)) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile |
| </p> |
| </td> |
| <td> |
| <p> |
| k = log1p(-x) / log1p(-p) -1 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile from the complement |
| </p> |
| </td> |
| <td> |
| <p> |
| k = log(x) / log1p(-p) -1 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mean |
| </p> |
| </td> |
| <td> |
| <p> |
| (1-p)/p |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| variance |
| </p> |
| </td> |
| <td> |
| <p> |
| (1-p)/p² |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mode |
| </p> |
| </td> |
| <td> |
| <p> |
| 0 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| skewness |
| </p> |
| </td> |
| <td> |
| <p> |
| (2-p)/√q |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis |
| </p> |
| </td> |
| <td> |
| <p> |
| 9+p²/q |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis excess |
| </p> |
| </td> |
| <td> |
| <p> |
| 6 +p²/q |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| parameter estimation member functions |
| </p> |
| </td> |
| <td> |
| <p> |
| See <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative |
| Binomial Distribution</a> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| <code class="computeroutput"><span class="identifier">find_lower_bound_on_p</span></code> |
| </p> |
| </td> |
| <td> |
| <p> |
| See <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative |
| Binomial Distribution</a> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| <code class="computeroutput"><span class="identifier">find_upper_bound_on_p</span></code> |
| </p> |
| </td> |
| <td> |
| <p> |
| See <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative |
| Binomial Distribution</a> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| <code class="computeroutput"><span class="identifier">find_minimum_number_of_trials</span></code> |
| </p> |
| </td> |
| <td> |
| <p> |
| See <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative |
| Binomial Distribution</a> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| <code class="computeroutput"><span class="identifier">find_maximum_number_of_trials</span></code> |
| </p> |
| </td> |
| <td> |
| <p> |
| See <a class="link" href="negative_binomial_dist.html" title="Negative Binomial Distribution">Negative |
| Binomial Distribution</a> |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </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> |
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