The random variables \(\xi_{1}, \xi_{2}, \ldots, \xi_{n}, \ldots\) are independent and uniformly distributed over \((0,1)\). Let \(v\) be a random variable equal to the \(k\) for which the sum

\[ s_{k}=\xi_{1}+\xi_{2}+\ldots+\xi_{n} \]

exceeds 1 for the first time. Prove that \(M v=e\).