The Gaussian Distribution¶
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gsl_ran_gaussian
(sigma)¶ This function returns a Gaussian random variate, with mean zero and standard deviation
sigma
. The probability distribution for Gaussian random variates is,\[p(x) dx = {1 \over \sqrt{2 \pi \sigma^2}} \exp (-x^2 / 2\sigma^2) dx\]for \(x\) in the range \(-\infty\) to \(+\infty\). Use the transformation \(z = \mu + x\) on the numbers returned by
gsl_ran_gaussian
to obtain a Gaussian distribution with mean \(\mu\). This function uses the Box-Muller algorithm which requires two calls to the random number generator.
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gsl_ran_gaussian_pdf
(x, sigma)¶ This function computes the probability density \(p(x)\) at \(x\) for a Gaussian distribution with standard deviation
sigma
, using the formula given above.
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gsl_ran_gaussian_ziggurat
(sigma)¶
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gsl_ran_gaussian_ratio_method
(sigma)¶ These functions compute a Gaussian random variate using the alternative Marsaglia-Tsang ziggurat and Kinderman-Monahan-Leva ratio methods. The Ziggurat algorithm is the fastest available algorithm in most cases.
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gsl_ran_ugaussian
()¶
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gsl_ran_ugaussian_pdf
(x)¶
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gsl_ran_ugaussian_ratio_method
()¶ These functions compute results for the unit Gaussian distribution. They are equivalent to the functions above with a standard deviation of one,
sigma
= 1.
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gsl_cdf_gaussian_P
(x, sigma)¶
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gsl_cdf_gaussian_Q
(x, sigma)¶
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gsl_cdf_gaussian_Pinv
(P, sigma)¶
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gsl_cdf_gaussian_Qinv
(Q, sigma)¶ These functions compute the cumulative distribution functions \(P(x), Q(x)\) and their inverses for the Gaussian distribution with standard deviation
sigma
.
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gsl_cdf_ugaussian_P
(x)¶
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gsl_cdf_ugaussian_Q
(x)¶
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gsl_cdf_ugaussian_Pinv
(P)¶
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gsl_cdf_ugaussian_Qinv
(Q)¶ These functions compute the cumulative distribution functions \(P(x), Q(x)\) and their inverses for the unit Gaussian distribution.