MTH 223 Tutorial 5
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Tutorial 5
MTH 223
1. Given Λ = λ , N has a Poisson distribution with mean λ . The mixing random variable Λ has a uniform distribution on the interval (0,5). Determine the unconditional probability that N ≥ 2.
2. Let N be the number of claims in an insurance portfolio. Assume that the conditional distribution of N , given Θ = θ , is a negative binomial NB(θ, 5). Θ has a uniform distribution U (0, 8).
(a) Calculate the expectation of the number of claims in the portfolio. (b) Calculate the variance of the number of claims in the portfolio.
(c) Calculate the probability that there are at least two claims in the portfolio.
3. Assume that the conditional distribution of X, given Θ = θ, is a Bernoulli distribution with mean θ . The distribution of Θ is a beta distribution BET(1, 3).
(a) Show that X has a Bernoulli distribution and identify the parameter for the Bernoulli distribution.
(b) Show that the conditional distribution of Θ, given X = 0, is a beta distribution and identify the parameters for the beta distribution.
4. Given Θ = θ , N has a Poisson distribution with mean θ . The random variable Θ has a p.d.f.
u(θ) = α (α + 1)2-1 (θ + 1)e-a9 , θ > 0.
(a) Determine the p.m.f. of the mixed distribution.
(b) Show that the mixed distribution is also a compound distribution. Identify the primary and secondary distributions.
Hint: In this part, the logarithmic distribution with parameter β, which has a p.m.f.
╱ 
、k
pk = k ln(1 + β) , k = 1, 2, . . .
may be involved. The logarithmic distribution has a p.g.f. as follows
ln (1 ← β(t ← 1))
ln(1 + β) .
5. It has been determined from the past studies that the number of workers’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 一 GAM(r, β), and suppose that, given Λ = λ , N 一 POI(λ + µ) where µ > 0 is a constant.
(a) Show that N has p.g.f.
GN(t) = eu(t-1) [1 ← β(t ← 1)]-r .
(b) Find p0 .
(c) Show that G
(t) = [µ + rβ(1 + β ← βt)-1] GN(t), and hence
(1 + β)G
(t) = βtG
(t) + [µ(1 + β) + rβ] GN(t) ← µβtGN (t).
(d) Use part (c) to show that
p1 = [µ + rβ/(1 + β)]p0
and
pn+1 = for n = 1, 2, … … … .
(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
o θne-9
pn = Pr(N = n) = g(θ)dθ for n = 0, 1, 2, … … … ,
1 2 − 入
g(θ) = β e- 礻 for θ > λ and λ, β > 0.
(a) Prove that
pn = 
入β j 
0 
╱ 
、j λn-j for n = 0, 1, 2, … … … .
(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
GN(t) = e入(t-1) [1 ← β(t ← 1)]-1 .
Explain in words what type of distribution this p.g.f. corresponds to.
(d) Determine ↓(N) and Var(N).
(e) Consider a ground up loss random variable X which has a lognormal 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.
2023-05-13