First, suppose that i.e. the loss from event i is fixed at x$_{i}$ . Then All of the bounds are trivial to compute. Second, suppose that each f$_{i}$ is a Gamma distribution with parameters ( α$_{i}$, β$_{i}$ ): for α$_{i}$, β$_{i}$ > 0, where 1 is the indicator function and Γ is the Gamma function, Then The moments are and hence f$_{i}$ ( x ) = δ ( x − x$_{i}$ ) i = 1 , . . . , m, E( X k i ) = x k i and M$_{X}$$_{i}$ ( v ) = e $^{vx$_{i}$}$. f$_{i}$ ( x ) = Gam( x ; α$_{i}$, β$_{i}$ ) = β α$_{i}$ i Γ( α$_{i}$ ) x α$_{i}$ − $^{1}$e − β$_{i}$$^{x}$1$_{x>}$$_{0}$ i = 1 , . . . , m Γ( s ) := ∫ ∞ 0 x s − $^{1}$e − $^{x}$d x. M$_{i}$ ( v ) = ( β$_{i}$ β$_{i}$ − v$^{)}$$^{α$_{i}$}$ 0 ≤ v < β$_{i}$. (4) E( X k i ) = Γ( α$_{i}$ + k ) β k i Γ( α$_{i}$ ) (5) E( X$_{i}$ ) = α$_{i}$ β$_{i}$ , E( X 2 i ) = ( α$_{i}$ + 1) α$_{i}$ β 2 i . 13