38. For an insurance:
(i) The number of losses per year has a Poisson distribution with 10 λ= .
(ii) Loss amounts are uniformly distributed on (0, 10).
(iii) Loss amounts and the number of losses are mutually independent.
(iv) There is an ordinary deductible of 4 per loss.
Calculate the variance of aggregate payments in a year.
(A) 36
(B) 48
(C) 72
(D) 96
(E) 120
Exam M: Fall 2005 -39- GO ON TO NEXT PAGE
39. For an insurance portfolio:
(i) The number of claims has the probability distribution
n n p
0 0.1
1 0.4
2 0.3
3 0.2
(ii) Each claim amount has a Poisson distribution with mean 3; and
(iii) The number of claims and claim amounts are mutually independent.
Calculate the variance of aggregate claims.
(A) 4.8
(B) 6.4
(C) 8.0
(D) 10.2
(E) 12.4
Exam M: Fall 2005 -40- STOP
40. Lucky Tom deposits the coins he finds on the way to work according to a Poisson process
with a mean of 22 deposits per month.
5% of the time, Tom deposits coins worth a total of 10.
15% of the time, Tom deposits coins worth a total of 5.
80% of the time, Tom deposits coins worth a total of 1.
The amounts deposited are independent, and are independent of the number of deposits.
Calculate the variance in the total of the monthly deposits.
(A) 180
(B) 210
(C) 240
(D) 270
(E) 300
**END OF EXAMINATION**