Exercise: Processes in the park

Exercises $$May $8,202419$:$59+08$

Exercise: Processes in the park

$2()$

As in an earlier exercise, busy people arrive at the park according to a Poisson process with rate ${}_1=3/$hour

and stay in the park for exactly $\frac{1}{6}$ of an hour. Relaxed people arrive at the park according to a Poisson

process with rate ${}_2=\frac{2}{?}$ hour and stay in the park for exactly half an hour. The arrivals of busy and relaxed

people are independent processes. Assume that no$\text{'}$ other people arrive at the park.

Is the process of total arrivals at the park a Poisson process? If yes, enter the rate of that process in the

answer box below. If it is not, enter $0.$

Whenever a relaxed person exits the park, he$/$she enters a nearby coffee shop. $($Assume$,$ for simplicity, that

going from the park to the coffee shop takes zero time.$)$

Is the process of arrivals of relaxed persons at the coffee shop a Poisson process? If yes, enter the rate of that

process in the answer box below. If it is not, enter $0.$