A brownian bridge evaluate at a particular random variable

Question

I was wondering of someone could help with the following. I have a random variable given as $\lambda^{*}=\arg \max_{\lambda \in (0,1)} [B(\lambda)-\lambda B(1)]^{2}/\lambda(1-\lambda)$. I am now interested in the behaviour of the normalised squared brownian bridge evaluated at $\lambda^{*}$ itself i.e. $Z=[B(\lambda^{*})-\lambda^{*} B(1)]^{2}/\lambda^{*}(1-\lambda^{*})$. Shouldn't this be a $\chi^{2}(1)$? - Thanks to all. -- John