 Solution

For $x>\mu$, the corresponding smaller root is ${\mu }^2/x$. The conditional probability of selecting the larger root isIt is the same weight as a function of the actual selected value $x$. HenceCombining the two open intervals proves the output has the Inverse Gaussian distribution $IG(\mu ,\lambda )$. The event $Y=0$ has probability zero, so no atom occurs at $\mu$; the limiting density there is finite. The algorithm always selects one of its positive roots, so these branch densities exhaust its probability distribution.