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Consider a system, where at the time of arrival of a tagged packet, it sees \(\displaystyle N_Q\) packets.

**Given that the tagged packet arrives at an instant \(\displaystyle t\), which is uniform in [0, T],**

what is the probability that all \(\displaystyle N_Q\) packets arrived in [0,t]?

what is the probability that all \(\displaystyle N_Q\) packets arrived in [0,t]?

This is how i approached.

\(\displaystyle P\{N_Q \text{arrivals happened in} (0,t) |t\}= \frac{(\lambda \tau)^N_Q e^{-\lambda }}{N_Q!}\)

unconditioning on t, we get \(\displaystyle \frac{1}{T} \int _0^{T}\frac{(\lambda t)^N_Q e^{-\lambda t}}{N_Q!}dt\)

Here \(\displaystyle N_Q\) is a random variable in itself.

How do we get the expression independent of \(\displaystyle N_Q\)?