Prove that as a consequence of the reflection principle (formula (3.1.1)), for any fixed \(t\) :

(i) \(2 M_{t}-W_{t}\) is distributed as \(\left\|B_{t}^{(3)}\right\|\) where \(B^{(3)}\) is a 3-dimensional \(\mathrm{BM}\), starting from 0 ,

(ii) given \(2 M_{t}-W_{t}=r\), both \(M_{t}\) and \(M_{t}-W_{t}\) are uniformly distributed on \([0, r]\).

This result is a small part of Pitman's theorem.

Formula 3.1.1:

\[\begin{equation*}
\mathbb{P}\left(W_{t} \leq x, M_{t} \geq y\right)=\mathbb{P}\left(W_{t} \geq 2 y-x\right) \tag{3.1.1}
\end{equation*}\]