= Distribution of an internally studentized Gaussian residual
{title2=$\eta_i^2/\nu\sim\operatorname{Beta}(1/2,(\nu-1)/2)$}
In a <normal linear model> with residual <degrees of freedom> $\nu\geq2$, an <internally studentized residual> satisfies $|\eta_i|\leq\sqrt\nu$ and $\eta_i^2/\nu\sim\operatorname{Beta}(1/2,(\nu-1)/2)$, provided its <leverage> is less than one. Choose an <orthonormal basis> of the residual space with its first direction proportional to the projection of the $i$th coordinate vector. The <Gaussian> coordinates in this basis are independent standard <normal random variables> $Z_1,\ldots,Z_\nu$, and $\eta_i=\sqrt\nu Z_1/(\sum_jZ_j^2)^{1/2}$. The <beta distribution> follows from the independent <chi-squared distributions> of $Z_1^2$ and $\sum_{j=2}^\nu Z_j^2$. The bounded residual is not exactly distributed according to <Student's t-distribution>; it approaches the standard <normal distribution> as $\nu\to\infty$.
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