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!set gl_author=Sophie, Lemaire
!set gl_keywords=continuous_probability_distribution
!set gl_title=
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<div class="wims_defn"><h4>Definition</h4>
Let \(n) be a positive integer. The <span class="wims_emph">Student distribution with \(n) degrees
of freedom </span> (denoted by
\(\mathcal{T }(n))) is the distribution of the following random variable
\(\frac{X}{\sqrt{Y/n}}) where \(X) is \(\mathcal{N}(0,1)) distributed and
\(Y) is independent of \(X) and \(\chi^{2}(n)) distributed.
It is a continuous distribution over \(\RR) with density function
<div class="wimscenter">
\(x\mapsto\frac{\Gamma(\frac{n+1}{2})}{\sqrt{n\pi}
\Gamma(\frac{n}{2})}(1+\frac{x^2}{n})^{-\frac{n+1}{2}})
</div>
</div>
<table class="wimsborder wimscenter">
<tr><th>Expectation</th><th>Variance</th><th>Characteristic function</th></tr>
<td>0 si \(n>1) </td><td>\(\frac{n}{n-2}) si \(n>2)</td><td></td></tr></table>
