Appendix C

Some Fundamentals of the Risk-Based Approach

BASIC PRINCIPLES

The fundamental tools needed for the quantitative risk-based approach to decision-making include the basic principles of probability. Those principles start with the premise that in the presence of uncertainty, a phenomenon or physical process can be defined or represented by a *random variable* and its *probability distribution*. That is, uncertainty is modeled as a random variable with a range of possible values and their probabilities defined by a probability distribution.

Thus, if *X* is a random variable with a range of possible values from *a* to *b*, its probability distribution may be defined as *F _{X}(x)=P(X≤x); a≤x≤b*.

Within the range of possible values of a random variable, there will be a *mean* (or average) value and a measure of dispersion, such as the *variance* or *standard deviation*. The ratio of the standard deviation to the mean is the *coefficient of variation* (COV).

Among the useful probability distributions are the *normal* or *Gaussian* distribution and the *lognormal* (or logarithmic normal) distribution.

*The Normal or Gaussian Distribution*. The normal distribution, whose range of possible values is −∞ to +∞ is denoted as *N*(µ, σ), where *µ* is its mean value and *σ* is its standard deviation. If *µ*= 0 and *σ*= 1.0, the distribution is called the *standard normal distribution*. For the standard normal distribution, the probability from −∞ to *x* is *F _{X}(x)= Φ(x)*, where Φ(

where *µ _{X}* and

*The Lognormal Distribution.* In the lognormal distribution, whose range of possible values is 0 to ∞, there are no negative values. The probability that *X* will be between *a* and *b* becomes