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First, suppose that 

i.e. the loss from event i is fixed at x$_{i}$ . Then 

All of the bounds are trivial to compute. 

<img src='content_image/316652.jpg'>

<img src='content_image/316653.jpg'>

Second, suppose that each f$_{i}$ is a Gamma distribution with parameters ( α$_{i}$, β$_{i}$ ): 

<img src='content_image/316654.jpg'>

for α$_{i}$, β$_{i}$ > 0, where 1 is the indicator function and Γ is the Gamma function, 

Then 

The moments are 

and hence 

<img src='content_image/316656.jpg'>

<img src='content_image/316655.jpg'>

<img src='content_image/316657.jpg'>

<img src='content_image/316658.jpg'>

