Example
Example 5.21 Insurance claims are made at times distributed according to a Poisson process with rate λ; the successive claim amounts are independent random variables having distribution G with mean μ, and are independent of the claim arrival times. Let Si and Ci denote, respectively, the time and the amount of the ith claim. The total discounted cost of all claims made up to time t, call it D(t), is defined by D(t) =
N(t)
e−αSi Ci
i=1
where α is the discount rate and N(t) is the number of claims made by time t. To determine the expected value of D(t), we condition on N(t) to obtain E[D(t)] =
∞ n=0
E[D(t)|N(t) = n]e−λt
(λt)n n!
5.3 The Poisson Process
335
Now, conditional on N(t) = n, the claim arrival times S1 , … , Sn are distributed as the ordered values U(1) , … , U(n) of n independent uniform (0, t) random variables U1 , … , Un . Therefore, , n −αU(i)