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<title>GNU Scientific Library &ndash; Reference Manual: The Bivariate Gaussian Distribution</title>

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<a name="The-Bivariate-Gaussian-Distribution"></a>
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<p>
Next: <a href="The-Multivariate-Gaussian-Distribution.html#The-Multivariate-Gaussian-Distribution" accesskey="n" rel="next">The Multivariate Gaussian Distribution</a>, Previous: <a href="The-Gaussian-Tail-Distribution.html#The-Gaussian-Tail-Distribution" accesskey="p" rel="previous">The Gaussian Tail Distribution</a>, Up: <a href="Random-Number-Distributions.html#Random-Number-Distributions" accesskey="u" rel="up">Random Number Distributions</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<a name="The-Bivariate-Gaussian-Distribution-1"></a>
<h3 class="section">20.4 The Bivariate Gaussian Distribution</h3>

<dl>
<dt><a name="index-gsl_005fran_005fbivariate_005fgaussian"></a>Function: <em>void</em> <strong>gsl_ran_bivariate_gaussian</strong> <em>(const gsl_rng * <var>r</var>, double <var>sigma_x</var>, double <var>sigma_y</var>, double <var>rho</var>, double * <var>x</var>, double * <var>y</var>)</em></dt>
<dd><a name="index-Bivariate-Gaussian-distribution"></a>
<a name="index-two-dimensional-Gaussian-distribution"></a>
<a name="index-Gaussian-distribution_002c-bivariate"></a>
<p>This function generates a pair of correlated Gaussian variates, with
mean zero, correlation coefficient <var>rho</var> and standard deviations
<var>sigma_x</var> and <var>sigma_y</var> in the <em>x</em> and <em>y</em> directions.
The probability distribution for bivariate Gaussian random variates is,
</p>
<div class="example">
<pre class="example">p(x,y) dx dy = {1 \over 2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp (-(x^2/\sigma_x^2 + y^2/\sigma_y^2 - 2 \rho x y/(\sigma_x\sigma_y))/2(1-\rho^2)) dx dy
</pre></div>

<p>for <em>x,y</em> in the range <em>-\infty</em> to <em>+\infty</em>.  The
correlation coefficient <var>rho</var> should lie between <em>1</em> and
<em>-1</em>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fran_005fbivariate_005fgaussian_005fpdf"></a>Function: <em>double</em> <strong>gsl_ran_bivariate_gaussian_pdf</strong> <em>(double <var>x</var>, double <var>y</var>, double <var>sigma_x</var>, double <var>sigma_y</var>, double <var>rho</var>)</em></dt>
<dd><p>This function computes the probability density <em>p(x,y)</em> at
(<var>x</var>,<var>y</var>) for a bivariate Gaussian distribution with standard
deviations <var>sigma_x</var>, <var>sigma_y</var> and correlation coefficient
<var>rho</var>, using the formula given above.
</p></dd></dl>

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