At time t 0 you begin counting the arrivals of buses at a

Question

At time t = 0, you begin counting the arrivals of buses at a depot. The number of buses K_i that arrive between time i - 1 minutes and time i minutes, has the Poisson PMF

And K₁, K₂,... are an iid random sequence. Let R_i = K₁ + K₂ +···+ K_i denote the number of buses arriving in the first i minutes.

(a) What is the moment generating function φ_Ki (s)?

(b) Find the MGF φ_Ri(s).

(c) Find the PMF P_Ri(r).

(d) Find E[R_i] and Var[R_i]