Poisson random variable on uniformly distributed variables

Question

Let $U_1, U_2, ...$ be $i.i.d$ random variables that are uniformly distributed on $(0,1)$. We define: $N((0,s]) = sup(k \in N: \sum_{i=1}^k (-log(1- U_i)) \leq s)$

Is $N((0,s])$ a Poisson variable with parameter $s$ ?

Answer 1

Yes. The $-\log(1-U_i)$ are independent standard exponentials. Since the waiting time of a Poisson process is exponential, your expression gives the number of clicks of a standard poisson process before time $s$, which is Poisson with mean $s.$