Non-homogeneous Poisson Process & P.G.F.

Hi, below are two questions which I have problem solving. Help is very much appreciated! Thanks in advanced!

**Q1**. During the night shift in an A&E department (10 pm to 6am) patients arrived at department requiring emergency treatment. The arrival of emergencies may be modeled as a non-homogeneous poisson process with hourly rate λ(t) = 2t/(1+t2), 0 ≤ t ≤ 8.

Find the expected number of emergencies between midnight and 2 am. Provide your answer in three decimal places.

**Q2**. Calls are made to a telephone helpline according to a poisson process at an average rate of 3 calls per hour. The number of questions asked by each caller has a geometric distribution (starting at 1) with mean 2.5.

Find the p.g.f. of the total number of questions asked in a period of three hours and use the p.g.f. to find variance of the number of questions asked in a period of three hours.