Let \(q(t)\) be the survival probability and let \(q^{-1}\) be its inverse function. Also, let \(U\) be a uniform random variable on \([0,1]\). For each realization \(u\), let \(\tau\) be chosen such that \(q(\tau)=u\). In this chain of relations, state whether each * should be \(\leq,=\), or \(\geq: \mathrm{P}[\tau * t]=\mathrm{P}\left[q^{-1}(U) * t\right]=\mathrm{P}[U * q(t)]\). Conclude that since \(U\) is uniform, \(\mathrm{P}[\tau \geq t]=q(t)\).