Expected value of series of uniformly converges random variables

Question

Let $X_1,X_2,X_3,...$ a series of i.i.d. variables with $X_i \sim \mathcal{U}(0,1)$.
Let $N=\inf\{n\mid \sum_{i=1}^{n}X_i\geq1\}$
Prove that $E(N)=e$.

I don't really have a clue how to even start proving that. Can someone please help?
Thanks.

Answer 1

Hints for the steps of a possible procedure:

1. Show that $$P(N>t) = P\left(\sum_{i=1}^t x_i<1\right)$$

2. Compute the above probability by geometric considerations (volume of a simplex)

3. Use the property that for a non-negative valued variable $Y$, $E(Y)=\sum_{t=1}^{\infty}{ P(Y\ge t)}$