# Poisson Processes

*You should solve these problems with as few calculations as possible, relying on properties of Poisson processes as much as possible.*

- Calls arrive at a customer service center according to a Poisson process with with \(\lambda=3\) calls per minute. Compute and interpret the following quantities.

- \(\text{E}(N_1)\).
- \(\text{SD}(N_1)\).
- \(\text{P}(N_1=2)\).
- \(\text{P}(N_1=2, N_3=6)\).
- \(\text{P}(N_1=2|N_3=6)\).
- \(\text{P}(N_3=6|N_1=2)\).
- \(\text{E}[N_{15}|N_{10}=8]\)
- \(\text{E}[N_{10}|N_{15}=8]\).
- \(\text{Cov}(N_{10}, N_{15})\).
- \(\text{Corr}(N_{10}, N_{15})\).

- Arrivals of spam emails to your email spam filter follow a Poisson process with mean rate 1.5 spam emails per minute. For the parts below in addition to computing, denote the corresponding probability in terms of proper symbols and notation.

- Compute the probability that exactly 4 spam emails arrive to the filter in the next 2 minutes.
- Compute the conditional probability that more than 3 minutes elapse, starting now, before the next spam email arrives, given that the most recent spam email arrived 2 minutes ago.
- Compute the probability that the next spam email arrives some time after 3 minutes but before 5 minutes from now.

- Compute the probability that exactly one spam email arrives in the time interval from 3 minutes to 5 minutes from now.
- If only 1 spam email arrives in the next 5 minutes, compute the conditional probability that it arrives in the next minute.
- Compute the conditional probability that 2 spam emails arrive in the first minute, given that 5 spam emails arrive in the first 3 minutes.