Formal–statistical methods - Probabilistic Safety Assessment for Optimum Nuclear Power Plant Life Management (PLiM) - page 66

The study of statistical data reveals the frequency of occurrence of
hazardous events.However, the seriousness of the events (even within
a single class of failures) may vary considerably from event to event;
consequently, it is necessary to introduce categories of events (for example,
events with severe, moderate or minor effects) and to consider the frequency
of each of these categories. The latter is achieved by allocating a risk
indicator (number of events over a certain period of time divided by the
duration of this period) to each class or subclass which has the dimension
of inverse time. This figure is sometimes seen as a measure of 'probability'
of the event.
For example, it is possible to characterise the phenomenon of random
variable z by the incidence of events (phenomenon of implementation) for
a certain period of time T , for example, for the year. It is well known that
the mathematical expectation Mz of the random variable z is the average
(expected) number of events in a year or the frequency of occurrence of the
events. Then, in accordance with the accepted terminology in mathematical
statistics, the number of events (which is taken from statistical data) is a
sample, the ratio of the number of events to the duration of the observation
period - statistics, which is obviously an unbiased and consistent estimator
of the mathematical expectation Mz or the frequency of occurrence of
events. For example, if we assume the Possion distribution of z , i.e. if we
put

= = ⋅ [2.6]
where r is a constant, it is possible to evaluate the condition when the input
parameter can be considered as reliability. In fact, for the Poisson distribution
Mz = r·T . On the other hand, for the Poisson distribution the probability that
at time T at least one event happens is 1- e -rT . Therefore, the input parameter
can be interpreted only at very low frequencies of an event as the probability
of occurrence of at least one event during time T .
However, it must be noted that the parameter inputted in this way is
not an indicator of the probability in the precise, mathematical sense of the
word. The probability of occurrence (of an event in the final scheme in the
classical sense) is the ratio of the power of the set of elementary outcomes
that make up this event to the power of the entire set of elementary
outcomes. The probability of an event is a real number lying in the range
0-1. Thus, in the case considered here the links between events A and B ,
where B occurs only if event A occurs, can be interpreted as the probability.
The risk can be defined quantitatively as the probability P of an event
when the event A occurs (a dimensionless quantity that lies within 0-1).
Since hazard is an accidental phenomenon, the hazard risk (no matter
how we define it - as frequency or probability) is a numerical characteristic
of the corresponding random variable used to describe this risk. As a simple
example of a possible formal approach consider the random variable s -
duration of trouble-free operation of an industrial enterprise whose domain

rT k
Pz k e rT k ( ) ( ) / !,

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