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 FX(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 FX(x)= Φ(x), where Φ(x) is tabulated in Tables of Standard Normal Probability. The probability of a random variable, X, between a and b can be evaluated as
where µX and σX are, respectively, the mean and standard deviation of X.
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