Third, suppose that each f$_{i}$ is a finite mixture of Gamma distributions: where ∑ p$_{i}$ k $_{=1}$π$_{ik}$ = 1 for each i . Then In other words, this is exactly the same as creating an extended ELT with plain Gamma f$_{i}$ ’s (i.e. as in the second case), but where each λ$_{i}$ is shared out among the p$_{i}$ mixture components according to the mixture weights π$_{i}$$_{1}$, . . . , π$_{ip}$$_{i}$ . This third case is very helpful, because the Gamma calculation is so simple, and yet it is possible to approximate any strictly positive absolutely continuous probability density function that has limit zero as x → ∞ , with a mixture of Gamma distributions (Wiper et al. , 2001). It is also possible to approximate point distributions by very concentrated Gamma distributions, discussed below in Section 6.3. Thus the secondary uncertainty for an event might be represented as a set of discrete losses, each with its own probability, but encoded as a set of highly concentrated Gamma distributions, leading to very efficient calculations. Capped single-event losses. For insurers, a rescaled Beta distribution is often preferred to a Gamma distribution, because it has a finite upper limit 14 f$_{i}$ ( x ) = p$_{i}$ ∑ k =1 π$_{ik}$ Gam( x ; α$_{ik}$, β$_{ik}$ ) i = 1 , . . . , m f$_{Y}$ ( y ) = m ∑ i =1 λ$_{i}$ λ p$_{i}$ ∑ k =1 π$_{ik}$ Gam( y ; α$_{ik}$, β$_{ik}$ ) = m ∑ i =1 p$_{i}$ ∑ k =1 λ$_{i}$π$_{ik}$ λ Gam( y ; α$_{ik}$, β$_{ik}$ ) .