Poissonized coupon completion time and count ($\lambda \mathbb{E}T=\mathbb{E}L$)

For independent coupon labels arriving at rate $\lambda$, Poisson thinning makes the type-specific first arrival times independent exponentials of rates $\lambda p_j$. Completion time is their maximum. Its distribution is $\mathbb{P}(T<t)={\prod }_j(1-e^{-\lambda p_{j}t})$. The completion count $L$ depends only on the labels and is independent of the exponential interarrival sequence. Therefore $T={\sum }_{k\ge 1}{E}_k{1}_{\{L\ge k\}}$ and nonnegative integration gives the displayed identity. Completion count and its expectation do not depend on arrival rate.