Distribution of the minimum number of uniform random variables that exceed a given number

Question

Let $X_i$, $i=1,2,3,...$ be an iid sequence of uniform random variables. I‘m interested in the distribution of $$Y_x=\inf\left\{k\geq0\,\middle|\,\sum_{i=1}^kX_i>x\right\}$$ for $x>0$. Does it have a name? What is known about it besides its mean?

Answer 1

Not a full answer, but if this helps:

$$P(Y_x > n) = P(\sum_{i=1}^n X_i \le x) = G(x;n)$$

where $$ G(x;n) = \frac{1}{n!} \sum_{k=0}^{\lfloor x \rfloor} (-1)^k \binom{n}{k} (x-k)^n$$

is the CDF of the Irwin–Hall distribution.