# Distributions for modelling operational risk capital

## Daniel Rodríguez

Probability distributions (French and Insua, 2000) play a role in the operational risk modelling process that permits us to interpolate and extrapolate from the observed loss sample. Indeed, in capital modelling, the capital charge is determined through probability distributions. The number of times an operational event takes place is determined by a discrete probability distribution, such as a Poisson or a negative binomial. On the other hand, the loss amount for any of the events is determined by a continuous positive probability distribution, such as gamma, lognormal or Pareto distributions. These examples are parametric distributions, where a mathematical expression and a set of parameters completely define the probability of all attainable values. It is also possible to define distributions using empirical data, mixing two or more parametric distributions or changing the range in which the distributions are defined by means of its truncation or shifting.

Mathematically, a probability distribution is a function that describes all the possible values and likelihoods that a random variable can take within a given range. This relationship can be expressed by different