Let \(\{X(t), t \geq 0\}\) be a death process with \(X(0)=n\) and positive death rates \(\mu_{1}, \mu_{2}, \ldots, \mu_{n}\).

Prove: If \(Y\) is an exponential random variable with parameter \(\lambda\) and independent of the death process, then

\[P(X(Y)=0)=\prod_{i=1}^{n} \frac{\mu_{i}}{\mu_{i}+\lambda}\]