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Branching process, Discretizing, Failure rate, Fitting, Geometric distribution, 60E05, 62E99, 60J80,

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The distribution of the sum of dependent risks is a crucial aspect in actuarial sciences, risk management and in many branches of applied probability. In this paper, we obtain analytic expressions for the probability density function (pdf) and the cumulative distribution function (cdf) of aggregated risks, modeled according to a mixture of exponential distributions. We first review the properties of the multivariate mixture of exponential distributions, to then obtain the analytical formulation for the pdf and the cdf for the aggregated distribution. We study in detail some specific families with Pareto (Sarabia et al, 2016), Gamma, Weibull and inverse Gaussian mixture of exponentials (Whitmore and Lee, 1991) claims. We also discuss briefly the computation of risk measures, formulas for the ruin probability (Albrecher et al., 2011) and the collective risk model. An extension of the basic model based on mixtures of gamma distributions is proposed, which is one of the suggested directions for future research. .es (V. Jordá). The modelLet Θ be a positive random variable with cdf F Θ (·) and Laplace-Stieltjes transform (hereinafter referred to as Laplace transform) L Θ (·), that is L Θ (s) = E[exp(−sΘ)] = ∞ 0 e −sz dF Θ (z). A distribution with support on (0, ∞) is identified by its Laplace transform. We consider the classical compound