MTH 223: Mathematical Risk Theory
Tutorial 3 Part I
1. For models involving general liability insurance, actuaries at the In-surance Services Office once considered a mixture of two Pareto distri-butions. They decided that five parameters were not necessary. The distribution they selected has c.d.f.
Note that the shape parameters in the two Pareto distributions differ by 2. The second distribution places more probability on smaller val-ues. This might be a model for frequent, small claims while the first distribution covers large, but infrequent claims.
Determine the mean and the second moment of the above two-point mixture distribution.
2. Let X be a random variable with a Pareto distribution PAR(α, θ). Let Y = ln(1 + X/θ). Determine the name of the distribution of Y and identify its parameters by looking up the Distribution Table.
3. Suppose that X | Λ has a Weibull survival function , x > 0, and Λ has an exponential distribution with mean of θ > 0. Demonstrate that the unconditional distribution of X is loglogistic with a pdf as follows
4. Suppose that X is a random variable with p.d.f.
Determine the p.d.f. and c.d.f. of .