MTH 223 Assignment 6
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Assignment 6
MTH 223
1. Suppose that M and N are two discrete counting r.v.’s. Consider the compound r.v. defined by S = Mi, where the Mi are i.i.d. as M, and independent of N . Compute gk = Pr(S = k) for k = 0, 1, 2, 3 under the following four primary distributions, each with secondary distribution given by
Pr(M = 0) = 0.1, Pr(M = 1) = 0.65, and Pr(M = 2) = 0.25.
(a) N is zero-truncated negative binomial with r = 3 and β = 4/3
(b) N is zero-modified Poisson with λ = 5 and p0(N) = 0.15.
2. Let S1 = Xi and S2 = 1 Yj, where N1 ~ POI(3), N2 ~ BIN(0.4, 5), and N1 is independent of N2 . Further assume that, given N1 and N2 , {Xi, Yi, i = 1, 2, . . .} are all independent of each other and moreover, Xi ~ U(0, 8) and Yj ~ EXP(5). Compute Cov(S1, S2 ).
3. In a compound r.v. S, the primary distribution is NB(β, r) with a positive integer r, and the secondary distribution is exponential with mean θ . Find an analytical expression for the c.d.f. of S .
4. A ground-up model of the losses has c.d.f.
1458000(45 + 2x)
The insurance policy calls for an ordinary deductible of 30 to be imposed. Moreover, the number of payments NP has a p.m.f.
pk = , with β = 7/4.
(a) On average, how many payments will be made on this policy?
(b) Show that the c.d.f. of YP, the per-payment random variable, is given by
13824000(105 + 2y)
2023-05-13