Prove that, in the case \(u=0\), setting \(\beta=((1-\alpha) / \alpha)^{1 / 2}\), and \(\Phi^{*}(x)=\sqrt{2 / \pi} \int_{x}^{\infty} e^{-y^{2} / 2} d y\)

\[\mathbb{P}(q(\alpha) \in d x)=\left\{\begin{array}{c}\sqrt{2 / \pi} e^{-x^{2} / 2} \Phi^{*}(\beta x) d x \text { for } x \geq 0 \\\sqrt{2 / \pi} e^{-x^{2} / 2} \Phi^{*}\left(-x \beta^{-1}\right) d x \text { for } x \leq 0\end{array} .\right.\]