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MTH 223: Mathematical Risk Theory
Tutorial 3 Part II
5. Suppose that loss X has a two-point mixture distribution with the following c.d.f.:
FX (x) = 0.3F1 (x) + 0.7F2 (x) for all x,
where Fj (x) is the c.d.f. of eXj for j = 1, 2, X1 has a normal distri- bution N(1.5, 2), and X2 has a gamma distribution GAM(2, 1/5), i.e. F1 (x) and F2 (x) are lognormal and loggamma distribution functions, respectively; see the Distribution Table for their expressions.
(a) Calculate the mean of the loss.
(b) Calculate the variance of the loss.
(c) Calculate the probability that the loss exceeds 10.
6. Assume that loss X has a three-component spliced distribution with following p.d.f.
(a) Calculate the mean of the loss.
(b) Calculate the variance of the loss.
(c) Calculate the distribution function FX (x) of the loss for all x ∈ (-∞, ∞).
(d) Calculate the probability that the loss is between 6 and 12 .
7. The conditional distribution of loss X , given Θ = θ > 0, is a uniform distribution U (0, θ). Θ has a Gamma distribution GAM (2, 6).
(a) Calculate the mean of the loss.
(b) Calculate the variance of the loss.
(c) Calculate the probability that the loss exceeds its mean.