= Uniform-sum crossing time below one
{title2=$\mathbb E N(x)=e^x\quad(0\leq x\leq1)$}
For independent <uniform distribution> increments on $(0,1)$ and $0\leq x\leq1$, let $N(x)$ be the first index at which their partial sum exceeds $x$. <First-step analysis> gives $M(x)=\mathbb E N(x)=1+\int_0^xM(s)ds$, hence $M=e^x$. Equivalently, $\mathbb P(N(x)>n)=x^n/n!$ is a simplex-volume probability and the <tail-sum formula> sums to $e^x$. The strict inequality gives $N(0)=1$ almost surely.
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