 Age of a Poisson process

With $J_0=0$, the age $A(t)=t-J_{N_t}$ of a rate-$\lambda$ Poisson process has meanGiven $N_t=n$, the last arrival is the maximum of $n$ uniform points and $\mathbb{E}[A(t)\mid N_t=n]=t/(n+1)$, including $n=0$.