Poisson Process

Claims against an insurance company follow a Poisson process with rate > 0. A total of N claims were received over two periods of combined length t = t₁ + t₂ with t₁ and t₂ being the lengths of the separate periods.

(a) Given this information, derive the (conditional) probability distribution of N₁, the number of claims made in the 1st period, given N.

(b) The amount paid for the ith claim is X_i , with X₁, X₂, . . . i.i.d. and independent of the claims process. Let E(X_i) = and Var(X_i) = 2 for i = 1, . . . , N. Given N, find the mean and variance of the total claims paid in period 1. That is, find these two conditional moments of the quantity:

i=1N1Xi

where by convention W₁ = 0 if N₁ = 0.