A store opens at 8am. From 8 until 10, customers arrive at a poisson rate of (4) per hour. Between 10 and 12, they arrive at a poisson rate of (8) per hour. From 12 to 2, the arrical rate increases steadily from (8) per hour at 12 to (10) per hour at 2. And from 2 to 5, the arrival rate drops steadily from (10) per hour 1t 2 to (4) per hour at 5. Determine the probability distribution of the number of customers that enter the store on a given day.

My approach:

Generating the intensity function, I have:

$\displaystyle \lambda(t)=4,(0<t<2)=8,(2<t<4)=t+4,(4<t<6)=22-2t,(6<t<9)$

Finding the respected M(t) (mean), I have 8,16,18,21. E.g. $\displaystyle \int_0^2 4.dt= 8 $

Can I then conclude that the distribution follows Poi(8+16+18+21)=Poi(63)?