Refer to the energy goodness-of-fit test of the two-parameter exponential family \(T \sim \operatorname{Exp}(\mu, \lambda)\) described in Section 3.1. Prove that under the null hypothesis, the expected distances are

\[
\begin{aligned}
& E|t-T|=t-\mu+\frac{1}{\lambda}\left(1-2 F_{T}(t)ight), \quad t \geq \mu \\
& E\left|T-T^{\prime}ight|=\frac{1}{\lambda}
\end{aligned}
\]