Possible Duplicate:
choose a random number between 0 and 1 and record its value. and keep doing it until the sum of the numbers exceeds 1. how many tries?
So I'm reading a book about simulation, and in one of the chapters about random numbers generation I found the following exercise:
For uniform $(0,1)$ random independent variables $U_1, U_2, \dots$ define
$$ N = \min \bigg \{ n : \sum_{i=1}^n U_i > 1 \bigg \} $$
Give an estimate for the value of $E[N]$.
That is: $N$ is equal to the number of random numbers uniformly distributed in $(0,1)$ that must be summed to exceed $1$. What's the expected value of $N$?
I wrote some code and I saw that the expected value of $N$ goes to $e = 2.71\dots$
The book does not ask for a formal proof of this fact, but now I'm curious!
So I would like to ask for
- A (possibily) simple (= undergraduate level) analytic proof of this fact
- An intuitive explanation for this fact
or both.