MTH 223: Mathematical Risk Theory

Tutorial 5 Part II

5. It has been determined from the past studies that the number of work-ers’ compensation claims for a group of 300 employees in a certain occupation class has the negative binomial distribution with β = 0.3 and r = 10. Determine the frequency distribution for a group of 500 such individuals.

6. Let Λ ∼ GAM(r, β), and suppose that, given Λ = λ, N ∼ POI(λ + µ) where µ > 0 is a constant.

(a) Show that N has p.g.f.

(b) Find p0.

(c) Show that

and hence

(d) Use part (c) to show that

and

(e) Assume that µ = 0.5, r = 3, and β = 2 for this part only, use parts (b) and (d) to calculate p0, p1, p2, p3 and p4.

7. Suppose that N has the following mixed Poisson distribution with pmf

where the mixing random variable Θ has p.d.f.

(a) Prove that

(b) Find the m.g.f. of Θ.

(c) Use the double expectation formula and your result in part (b) to show that the p.g.f. of N is given by

Explain in words what type of distribution this p.g.f. corresponds to.

(d) Determine E(N) and Var(N).

(e) Consider a ground up loss random variable X which has a lognor-mal distribution with parameters µ = 3.3 and σ = 2.5. Suppose a franchise deductible of 200 is imposed. If the number of losses has the p.g.f. in part (c), identify the distribution of the number of payments.