Question: Students arrive independently at the university central library. The number of students arriving in a given interval follows a Poisson distribution with a mean arrival rate of 15 students per hour.

1. What is the distribution of the inter-arrival process?
2. What is the probability that 30 students will arrive between \(1 \mathrm{pm}\) and \(3 \mathrm{pm}\) ?
3. If no students have arrived by \(1: 30 \mathrm{pm}\), what is the probability that 10 will arrive between \(2 \mathrm{pm}\) and \(3 \mathrm{pm}\) ?

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