= Poissonized coupon completion time and count
{title2=$\lambda\mathbb ET=\mathbb EL$}
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 $\mathbb P(T<t)=\prod_j(1-e^{-\lambda p_jt})$. The completion count $L$ depends only on the labels and is independent of the exponential interarrival sequence. Therefore $T=\sum_{k\ge1}E_k\mathbf1_{\{L\ge k\}}$ and nonnegative integration gives the displayed identity. Completion count and its expectation do not depend on arrival rate.
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