first is that the m events with different IDs follow concurrent but indepen- dent homogeneous Poisson processes. The second is that the collective of events follows a single homogeneous Poisson process with arrival rate and then, when an event occurs, its ID is selected independently at random with probability λ$_{i}$/λ . The second approach is more tractable for our purposes. Therefore we define Y as the loss incurred by a randomly selected event, with probability density function The total loss incurred over an interval of length t is then modelled as the random sum of independent losses, or The total loss S$_{t}$ would generally be termed a compound Poisson process with rate λ and component distribution f$_{Y}$ . An unusual feature of loss mod- elling is that the component distribution f$_{Y}$ is itself a mixture, sometimes with thousands of components. S$_{t}$ := N$_{t}$ ∑ j =1 Y$_{j}$ where      N$_{t}$ ∼ Poisson( λt ), and Y$_{1}$, Y$_{2}$, . . . iid ∼ f$_{Y}$ . λ := m ∑ i =1 λ$_{i}$ f$_{Y}$ = m ∑ i =1 λ$_{i}$ λ f$_{i}$ . 4