Sum of Uniform Variables

Let $X_1, X_2, \dots$ be a sequence of independent and identically distributed random variables, each following the uniform distribution $\mathcal{U}[0,1]$ .

Define:

N := \min \left\{ n \geq 1 \mid \sum_{i=1}^n X_i > 1 \right\}

In other words, $N$ is the number of draws needed for the sum to exceed 1.

Question: Determine the expectation $\mathbb{E}[N]$ .

Sum of Uniform Variables

Answer