A new distribution related to the logistic (Q1965941)

The author proposes a distribution of a random variable \(X_{\alpha}\) such that if \(Y\) is a standard logistic variable then \(X_{\alpha}+\alpha Y\) is distributed like \(Y\). This random variable \(X_{\alpha}\) can be simulated by transforming a standard uniform variable \(U\) as following: \(X_{\alpha}=\ln[ (\sin(\alpha\pi U)/ \sin(\alpha\pi(1-U))]\). As \(\alpha\to 0\), the distribution of \(X_{\alpha}\) approaches the standard logistic; as \(\alpha\to 1\), the distribution of \(X_{\alpha}/\pi(1-\alpha)\) approaches the standard Cauchy. If \(Y_{\alpha}\) and \(Y'_{\alpha}\) are independent each being positive stable with index \(\alpha\), then \(X_{\alpha}\) has the same distribution as \(\alpha\ln (Y_{\alpha}/Y'_{\alpha})\).