Does this distribution have a name?

Question

Independently select real numbers from a uniform distribution on the interval (0,1) until the sum of the selected numbers exceeds 1. Let Y be the random variable equaling the number of selections needed to exceed 1. The random variable Y is discrete with support {2,3,...}. The expectation of Y is e = 2.717... I believe that the probability that Y = k is (k-1)/k!. Is this correct? I also found some other interesting properties of the moment generating function for Y. Does this distribution have a name. Where might I find some information about this distribution so I can verify my findings

Answer 1

You should look here: https://en.wikipedia.org/wiki/Renewal_theory

In fact in your case:

$S_i \sim \mathcal{U} (0,1)$ i.i.d. , $J_n := \sum_{i=1}^n S_i$. Define

$$X_t := \sum_{n=1}^\infty 1_{\{ J_n \leq t \}} = \sup \{ n : J_n \leq t \}$$

Your special case: $Y = X_1 + 1$. (Note that $Y$ is a stopping-time for $\sigma (S_1 , \ldots , S_n)$; makes some things easier; for example you can use Wald's identity, to calculate the expectation value)

I think the case of $S_i$ uniform distributed should be a well-analysed case. There are several books about this topic.