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<a name="The-histogram-probability-distribution-struct"></a>
<div class="header">
<p>
Next: <a href="Example-programs-for-histograms.html#Example-programs-for-histograms" accesskey="n" rel="next">Example programs for histograms</a>, Previous: <a href="Resampling-from-histograms.html#Resampling-from-histograms" accesskey="p" rel="previous">Resampling from histograms</a>, Up: <a href="Histograms.html#Histograms" accesskey="u" rel="up">Histograms</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="The-histogram-probability-distribution-struct-1"></a>
<h3 class="section">23.10 The histogram probability distribution struct</h3>
<a name="index-probability-distribution_002c-from-histogram"></a>
<a name="index-sampling-from-histograms-1"></a>
<a name="index-random-sampling-from-histograms"></a>
<a name="index-histograms_002c-random-sampling-from"></a>
<p>The probability distribution function for a histogram consists of a set
of <em>bins</em> which measure the probability of an event falling into a
given range of a continuous variable <em>x</em>. A probability
distribution function is defined by the following struct, which actually
stores the cumulative probability distribution function.  This is the
natural quantity for generating samples via the inverse transform
method, because there is a one-to-one mapping between the cumulative
probability distribution and the range [0,1].  It can be shown that by
taking a uniform random number in this range and finding its
corresponding coordinate in the cumulative probability distribution we
obtain samples with the desired probability distribution.
</p>
<dl>
<dt><a name="index-gsl_005fhistogram_005fpdf"></a>Data Type: <strong>gsl_histogram_pdf</strong></dt>
<dd><dl compact="compact">
<dt><code>size_t n</code></dt>
<dd><p>This is the number of bins used to approximate the probability
distribution function. 
</p></dd>
<dt><code>double * range</code></dt>
<dd><p>The ranges of the bins are stored in an array of <em><var>n</var>+1</em> elements
pointed to by <var>range</var>.
</p></dd>
<dt><code>double * sum</code></dt>
<dd><p>The cumulative probability for the bins is stored in an array of
<var>n</var> elements pointed to by <var>sum</var>.
</p></dd>
</dl>
</dd></dl>

<p>The following functions allow you to create a <code>gsl_histogram_pdf</code>
struct which represents this probability distribution and generate
random samples from it.
</p>
<dl>
<dt><a name="index-gsl_005fhistogram_005fpdf_005falloc"></a>Function: <em>gsl_histogram_pdf *</em> <strong>gsl_histogram_pdf_alloc</strong> <em>(size_t <var>n</var>)</em></dt>
<dd><p>This function allocates memory for a probability distribution with
<var>n</var> bins and returns a pointer to a newly initialized
<code>gsl_histogram_pdf</code> struct. If insufficient memory is available a
null pointer is returned and the error handler is invoked with an error
code of <code>GSL_ENOMEM</code>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fhistogram_005fpdf_005finit"></a>Function: <em>int</em> <strong>gsl_histogram_pdf_init</strong> <em>(gsl_histogram_pdf * <var>p</var>, const gsl_histogram * <var>h</var>)</em></dt>
<dd><p>This function initializes the probability distribution <var>p</var> with
the contents of the histogram <var>h</var>. If any of the bins of <var>h</var> are
negative then the error handler is invoked with an error code of
<code>GSL_EDOM</code> because a probability distribution cannot contain
negative values.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fhistogram_005fpdf_005ffree"></a>Function: <em>void</em> <strong>gsl_histogram_pdf_free</strong> <em>(gsl_histogram_pdf * <var>p</var>)</em></dt>
<dd><p>This function frees the probability distribution function <var>p</var> and
all of the memory associated with it.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fhistogram_005fpdf_005fsample"></a>Function: <em>double</em> <strong>gsl_histogram_pdf_sample</strong> <em>(const gsl_histogram_pdf * <var>p</var>, double <var>r</var>)</em></dt>
<dd><p>This function uses <var>r</var>, a uniform random number between zero and
one, to compute a single random sample from the probability distribution
<var>p</var>.  The algorithm used to compute the sample <em>s</em> is given by
the following formula,
</p>
<div class="example">
<pre class="example">s = range[i] + delta * (range[i+1] - range[i])
</pre></div>

<p>where <em>i</em> is the index which satisfies 
<em>sum[i] &lt;=  r &lt; sum[i+1]</em> and 
<em>delta</em> is 
<em>(r - sum[i])/(sum[i+1] - sum[i])</em>.
</p></dd></dl>

<hr>
<div class="header">
<p>
Next: <a href="Example-programs-for-histograms.html#Example-programs-for-histograms" accesskey="n" rel="next">Example programs for histograms</a>, Previous: <a href="Resampling-from-histograms.html#Resampling-from-histograms" accesskey="p" rel="previous">Resampling from histograms</a>, Up: <a href="Histograms.html#Histograms" accesskey="u" rel="up">Histograms</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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