If a random variable x is equally likely to take any value in the interval 0 to 1, then its probability distribution function (PDF) is constant over this range. The beta PDF is a very flexible function for use in describing empirical data such as a’ and b’ as in Chap. 4. The general form of this distribution is given by Benjamin and Cornell (1970) as
f (x) = 1 xr 1(1 – x )r – 1 (0 <y< 1),
P
where в is the normalizing constant as
(r – 1)!(t – r – 1)!
ЇЇ-Т)
for integer r and t – r values, otherwise
Г(г)T(t – r)
r(t)
in which Г (.) is the incomplete gamma function of the argument. Herein, r and t are the PDF parameters related to the mean x and variance a2 parameters as follows:
_ r
x =
t
and
2 r (t – r ) ax — .
x t2(t + r)
