Suppose \(U_1,U_2, …\) are independent uniform \( (0,1) \) random variables. Let \(N\) be the first point of increase. That is to say the first \(n \geq 2\) such that \(U_n > U_{n-1}\). Show that for \(u \in (0,1)\):

- \[\mathbf{P}(U_1 \leq u \ { and } \ N=n)= \frac{u^{n-1}}{(n-1)!}-\frac{u^{n}}{n!} \quad;\quad n \geq 2\]
- \( \mathbf{E}(N)=e \)

Some useful observations:

- \[\mathbf{P}(U_1 \leq u \ { and } \ N=n) = \mathbf{P}(U_1 \leq u \ { and } \ N \geq n) -\mathbf{P}(U_1 \leq u \ { and } \ N \geq n+1)\]
- The following events are equal

\[ \{U_1 \leq u \quad{ and } \quad N \geq n\} = \{U_{n-1}\leq U_{n-2} \leq \cdots \leq U_2\leq U_{1} \leq u \}\] - \[ \mathbf{P}\{U_2 \leq U_1 \leq u \}= \int_0^u \int_0^{u_1} du_2 du_1 \]