Siegel median bound: $F(1)\ge\tfrac12$ for the homogeneous waiting time

Siegel median bound for the homogeneous waiting-time CDF at $t=1$. Consider the order-statistic waiting time $T$ where $N$ independent rate-$\lambda$ exponential clocks fire, with $\lambda=\log(N/m)$ (so $e^{-\lambda}=m/N$) and we wait for the $(N-m)$-th clock; its CDF is $F(s)=\sum_{k=N-m}^{N}\binom Nk(1-e^{-\lambda s})^k(e^{-\lambda s})^{N-k}$. The claim is $F(1)\ge \tfrac12$ for $1\le m<N$. This is the heart of the integer-mean binomial median theorem (Kaas–Buhrman / Jogdeo–Samuels): via Siegel's symmetrized-CDF (moustache) argument, the mean $\mu=\mathbb E[T]=\tfrac1\lambda\sum_{j=m+1}^N\tfrac1j<1$ satisfies $F(\mu)\ge\tfrac12$ (median $\le$ mean), and monotonicity of $F$ with $\mu<1$ gives $F(1)\ge F(\mu)\ge\tfrac12$. Source: Siegel, Median Bounds and their Application, J. Algorithms 38 (2001), Thm 2.1 (moustache value bound) and Thm 2.2 (homogeneous waiting-time model); equivalently Jogdeo–Samuels (Ann. Math. Statist. 39, 1968).