First ascent in independent continuous observations ($P(N=r)=(r-1)/r!,\mathbb{E}N=e$)

For independent and identically distributed random variables with a continuous probability distribution, let $N\ge 2$ be the index of the first increase. The event $N\ge n$ requires the first $n-1$ observations to be decreasing; symmetry among their $(n-1)!$ possible strict orders gives probability $1/(n-1)!$ for $n\ge 2$. Subtracting successive tails gives the displayed mass function for $r\ge 2$. The tails tend to zero, so $N$ is finite almost surely. The tail-sum formula for expectation gives $\mathbb{E}N=1+{\sum }_{j\ge 1}1/j!=e$, independently of the continuous sampling law.