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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) =-,

College Algebra
College Algebra
1st Edition
ISBN:9781938168383
Author:Jay Abramson
Publisher:Jay Abramson
Chapter6: Exponential And Logarithmic Functions
Section6.8: Fitting Exponential Models To Data
Problem 3TI: Table 6 shows the population, in thousands, of harbor seals in the Wadden Sea over the years 1997 to...
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Suppose that each individual in a large insurance portfolio incurs losses according to an exponential distribution with mean 1/2,where i varies over the portfolio according to a G(a, 3) mixing distribution. The respective densities of the two distributions are given by Sx (x) = (1/2) exp (-x/à), x > 0, 2 > 0; S(2) =- T(a) 2-l exp(-8i), 2 > 0. Given that the Pareto pdf given by fx (x) = (5 +x)ª+1 * > 0, a > 0, 8> 0. (a) Show that the marginal distribution of losses follows a Pareto distribution, i.e. P(a, 5). (b) Use the mixing formulation of the Pareto to deduce that if X~P(a, 5), then E (X) = .
Transcribed Image Text:Suppose that each individual in a large insurance portfolio incurs losses according to an exponential distribution with mean 1/2,where i varies over the portfolio according to a G(a, 3) mixing distribution. The respective densities of the two distributions are given by Sx (x) = (1/2) exp (-x/à), x > 0, 2 > 0; S(2) =- T(a) 2-l exp(-8i), 2 > 0. Given that the Pareto pdf given by fx (x) = (5 +x)ª+1 * > 0, a > 0, 8> 0. (a) Show that the marginal distribution of losses follows a Pareto distribution, i.e. P(a, 5). (b) Use the mixing formulation of the Pareto to deduce that if X~P(a, 5), then E (X) = .
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