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representing the maximum insured loss. The moment generating function of a Beta distribution is an untabulated function with an infinite series representation, and so will be more expensive to compute accurately; this will affect the Chernoff bound. There are no difficulties with the moments. 

However, we would question the suitability of using a Beta distribution here. The insurer’s loss from an event is capped at the maximum insured loss. This implies an atom of probability at the maximum insured loss: if f$_{i}$ is the original loss distribution for event i and u is the maximum insured loss, then 

<img src='content_image/307000.jpg'>

where p$_{i}$ := ∫ u 0 f$_{i}$ ( x ) d x and δ is the Dirac delta function, as before. A Beta distribution scaled to [0 , u ] would be quite different, having no atom at u . 

The Gamma distribution for f$_{i}$ is tractable with a cap on losses. If f$_{i}$ is a Gamma distribution then the MGF is 

where γ is the incomplete Gamma function, 

and 

<img src='content_image/307001.jpg'>

<img src='content_image/307002.jpg'>

<img src='content_image/307003.jpg'>

