provides a quantitative definition of risk (Kaplan and Garrick, 1981; ASME/ANS, 2009; IPET 2009) in probabilistic terms:
where v = frequency of occurrence or exceedance, and is a measure of the aleatory uncertainty (the randomness of events); ρ = probability as a measure of the confidence to which an estimate of v is the true value, or the epistemic uncertainty in the estimate of v. For purposes of a flood risk analysis, the consequences of flooding are measured in terms of the economic damages. Typically, results of risk analysis would be expressed in terms of a frequency distribution on economic damages (dollars). This is denoted
where n() = frequency of occurrence per year; and c = consequences measured in dollars.
Another risk metric is the expected annual losses, which can be estimated by
The estimate of flood consequences and their frequency of occurrence are uncertain, that is v(c) cannot be determined with certainty; there is epistemic in estimating the frequency and magnitude of floods, uncertainty in the performance of flood protection systems, and the magnitude of consequences. There is therefore a distribution on the estimate of v(c). A formal analysis of the uncertainties in the components of a risk analysis (h, v, and C) produces a probability distribution on the estimate of v(c), the estimate of risk. This probability distribution that quantifies the uncertainty in the estimate of v(c) can be thought of as a distribution from which confidence intervals on the estimate of risk can be derived.3 Kaplan and Garrick (1981) provide a quantitative definition of risk that includes the quantification of aleatory and epistemic uncertainties and the estimate of consequences:
or, in terms of the risk example above, equation (I-5) can be written,
where, v = frequency of occurrence or exceedence, as is a measure of the aleatory uncertainty; C = a consequence metric (e.g., economic impact, public safety); ρ = probability as a measure of epistemic uncertainty in the estimate; and where,
The frequency of occurrence is a measure of the aleatory uncertainty of events or outcomes (consequences), their randomness or stochastic character. An example of aleatory uncertainty is the occurrence of flood events in a watershed, or the performance of a levee for a given flood level. Equation (I-6) defines a discrete probability mass function on the frequency distribution of consequences, where probability is a measure of epistemic uncertainty in the estimate of v.
3 The notion of confidence intervals is a useful concept for purposes of illustrating the measure of uncertainty in the risk analysis results. In fact, however the assessment of uncertainties is not same as a statistical analysis of a dataset from which confidence intervals on parameter estimates are derived.