On Beta Distributions (Part 1)

In geophysical catastrophe modelling, the Beta distribution is the most popular probability distribution used to describe uncertainty in loss severity. The distribution is a continuous two-parameter function bounded by values of zero and unity.

The PDF of the Beta distribution is a function of its two shape parameters A and B:

P(x) = \beta(A,B)x^{A-1}(1-x)^{B-1}

β(A,B) simply normalises the distribution and is given by

\beta(A,B) = \cfrac{\Gamma(A+B)}{\Gamma(A)\Gamma(B)}

where Γ is the Gamma function.

The expected loss of the distribution is given by:

\mu = \cfrac{A}{A+B}

and its variance by:

\sigma^2 = \cfrac{AB}{(A+B)^2(A+B+1)}

In catastrophe modelling applications it is normal to want to derive the shape parameters from an expected loss and its standard deviation. A little work shows us that

A = \left(\left(\cfrac{\mu}{\sigma}\right)^2\left(1-\mu\right)\right)-\mu

B = A\left(\cfrac{1}{\mu}-1\right)

Note, the distribution is evaluated on values of x bounded by 0 and 1. In catastrophe modelling applications, the expected loss μ is represented by the ratio of the expected loss to the sum insured (or limit, or “exposed value”)

Author: admin

Working in cat for 20 years. Thinking about the universe for longer. Watching MotoGP since before I was born...