PROCEEDINGS May 15, 16, 17, 18, 2005 - Casualty Actuarial Society

60 RISKINESS LEVERAGE MODELSThe density function f(x) for the total sum of variables canmost easily be written asZ à !nXf(x)= dF± x ¡ x kZ´k=1dx 1 :::dx n f(x 1 ,:::,x n )±Ãx ¡nXx k!: (A.4)For calculation this is often a convenient form, as in thederivation of Equation (2.7):Z à nX! à nX!dF x i ¡ ¹ g kk=1k=1xk=1Z=Z=ZdxdF±Ãx ¡nXx k!(x ¡ ¹)g(x)k=1f(x)(x ¡ ¹)g(x)dx:(A.5)andSimilarly, the marginal density for any variable can be writtenZf k (y)= dF±(y ¡ x k ):(A.6)The cumulative distribution function for the total isZ à !nXF(x)= dF x ¡ x kZ´k=1dx 1 :::dx n f(x 1 ,:::,x n )f(x)= d dx F(x)emerges from simple differentiation rules.Ãx ¡nXx k!, (A.7)k=1(A.8)