Suppose that each individual in a large insurance portfolio incurs losses according to an exponential distribution with mean 1/2,where à varies over the portfolio according to a G(a,5) mixing distribution. The respective densities of the two distributions are given by fx (x) = (1/2) exp (-x/¿), x>0, 2 > 0; S(1) = -2ª-' exp(-di), i > 0. Га) ao Given that the Pareto pdf given by fx (x) = (5+x)e+1 * > 0, a > 0, 8> 0, (a) Show that the marginal distribution of losses follows a Pareto distribution, i.e. P(a,3). (b) Use the mixing formulation of the Pareto to deduce that if X~P(a, d), then E (X) =-,
Suppose that each individual in a large insurance portfolio incurs losses according to an exponential distribution with mean 1/2,where à varies over the portfolio according to a G(a,5) mixing distribution. The respective densities of the two distributions are given by fx (x) = (1/2) exp (-x/¿), x>0, 2 > 0; S(1) = -2ª-' exp(-di), i > 0. Га) ao Given that the Pareto pdf given by fx (x) = (5+x)e+1 * > 0, a > 0, 8> 0, (a) Show that the marginal distribution of losses follows a Pareto distribution, i.e. P(a,3). (b) Use the mixing formulation of the Pareto to deduce that if X~P(a, d), then E (X) =-,
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section: Chapter Questions
Problem 9T
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