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Result R3873 on D3353: Stopping time

Expectation of stopping time for I.I.D. sum of standard uniform random basic real numbers

Formulation 1

Let

$U_1, U_2, U_3, \ldots \in \text{Uniform}[0, 1]$ each be a D4624: Standard unsigned uniform random real number such that

(i)	$U_1, U_2, U_3, \ldots$ is an D2713: Independent random collection
(ii)	$x \in [0, \infty)$ is an D4767: Unsigned real number
(iii)	$\begin{equation} T_x : = \min \left\{ N \in \{ 1, 2, 3, \ldots \} : \sum_{n = 1}^N U_n > x \right\} \end{equation}$

Then

$\begin{equation} \mathbb{E} T_x = e^x \end{equation}$

Also known as

Expectation of first-passage time for I.I.D. sum of standard uniform random basic real numbers