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In page Student's t-distribution:

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There are various approaches to constructing random samples from the Student's t distribution. The matter depends on whether the samples are required on a stand-alone basis, or are to be constructed by application of a quantile function to uniform samples; e.g., in the multi-dimensional applications basis of copula-dependency.[citation needed] In the case of stand-alone sampling, an extension of the Box–Muller method and its polar form is easily deployed.[1] It has the merit that it applies equally well to all real positive degrees of freedom, ν, while many other candidate methods fail if ν is close to zero.[1]