= Independent exponential order-statistic spacings
{title2=$M_i-M_{i-1}\sim\operatorname{Exp}((n-i+1)\lambda)$}
For $n$ <independent and identically distributed> rate-$\lambda$ <exponential random variables>, let $M_0=0$ and $M_1<\cdots<M_n$ be their <order statistics>. The gaps $D_i=M_i-M_{i-1}$ are <independent> and have rates $(n-i+1)\lambda$. After the first clock rings, the <memoryless property> makes the surviving clocks fresh independent exponentials, independently of the elapsed minimum and the winning label. Induction gives the result. Equivalently, the ordered <joint probability density> is $n!\lambda^n\exp[-\lambda\sum_i m_i]$ on $0<m_1<\cdots<m_n$; changing to gaps gives a product of exponential densities because $\sum_i m_i=\sum_i(n-i+1)D_i$.
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