EXISTING MATHEMATICAL METHODS FOR UNCERTAINTY ASSESSMENT - Modelling Uncertainty in Flood Forecasting Systems - page 39

In case of discrete events, with X ={ x 1 ,…, x n }, the probability of the fuzzy event on

X can be expressed as:

(3.30)

Since this is a summation of the probability, p ( x = x i ), multiplied by the degree to which x i
belong to the fuzzy event it can be seen that it is a concept that is easy to accept
intuitively as the probability that the event will occur (Terano et al., 1992). The
concept of fuzzy probability is distinct from that of second-order probability (i.e. a
probability-value which is characterised by its probability distribution) and contains that
of interval-valued probability as a special case (Zadeh, 1984).

3.4.2 The concept of fuzzy-random variable

The term fuzzy-random variable was used by Ayyub and Chao (1998) to define an
uncertainty parameter that carries uncertainty from both fuzziness and randomness. The
method proposed by them to treat a fuzzy-random variable considers the membership
function as a weight function to the probability distribution function. If the uncertainty is
only due to randomness, (that is, if the variable is purely random) both the mean and the
standard deviation of its PDF are defined by crisp numbers. In contrast, for a fuzzy-
random variable either mean or standard deviation or both can be considered as a fuzzy
number defined by respective membership functions. The MF transferred to (equivalent)
PDF is used as a weight function to the original PDF. To explain this procedure, let us
consider a fuzzy-random variable x defined by its PDF, p X (x), with the mean and
standard deviation $. Let us assume that the standard deviation is a crisp number but the
mean is a fuzzy number defined by its membership function

The MF is then

transferred to a PDF

such that

(3.31)

where and are the upper bound and lower bound of the MF of the fuzzy mean
at support . The transformation used here is the same as the transformation by simple
normalisation as explained in Subsection 3.5.1. For convenience, the PDF is defined to be
a conditional PDF, Thus, if p X (x) is normal the expression for the

conditional PDF of p X (x) is given by