# J

where is the i-th segment of the damage.

In the PMRM, the concept ofthe expected value of damage is extended to generate multiple conditional expected-value functions, each associated with a particular range of exceedance probabilities or their corresponding range of damage severities. The resulting conditional expected-value functions, in conjunction with the traditional expected value, provide a family of risk measures associated with a particuiar policy.

Let 1 - on and 1 - a2, where 0 < a, < a2 < 1, denote exceedance probabilities that partition the domain of X into three ranges, as follows. On a plot of exceedance probability, there is a unique damage ft on the damage axis that corresponds to the exceedance probability 1 - a, on the probability axis. Similarly, there is a unique damage ft that corresponds to the exceedance probability 1 ~a2. Damages less than ft are considered to be oflow severity, and damages greater than ft of high severity. Similarly, damages of a magnitude between ft and ft are considered to be of moderate severity. The partitioning of risk into three severity ranges is illustrated in Fig. 7.2. For example, if the partitioning probability aj is specified to be 0.05, then ft is the 5th exceedance percentile. Similarly, if is 0.95 (i.e., 1- a? is equal to 0.05), then ft is the 95th exceedance percentile.

For each ofthe three ranges, the conditional expected damage (given that the damage is within that particular range) provides a measure ofthe risk associated with the range. These measures are obtained by defining the ccnditional expected value. Consequently, the new measures of risk 'dJe fi(-). of high exceedance probability and low severity;/3{-), of medium exceedance probability and moderate severity; andf4( ), oflow exceedance probability and high severity. The function/? (•) is the conditional expected value of X, given that x is less than or equal to ft:

Low severity

Low severity Damage X

Fig. 7.2 PDF of failure rate distributions for four designs.

Damage X

Fig. 7.2 PDF of failure rate distributions for four designs.

*P<*) dx M'} ~ />>(*) d*

rr i &xP(x)dx (77)

iTxpWdx f00

L pU)dx Jo

Thus, for a particular policy option, there are three measures of risk/?(■), fi(-), and^(-), in addition to the traditional expected value denoted by/5{-). The function/iG) is reserved for the cost associated with the management of risk. Note that