Probability theory, continuous random variables

In most natural situations, physical and chemical parameters are not defined by a unique deterministic value. Due to our limited comprehension of the natural processes and imperfect analytical procedures (notwithstanding the interaction of the measurement itself with the process investigated), measurements of concentrations, isotopic ratios and other geochemical parameters must be considered as samples taken from an infinite reservoir or population of attainable values. Defining random variables in a rigorous way would require a rather lengthy development of probability spaces and the measure theory which is beyond the scope of this book. For that purpose, the reader is referred to any of the many excellent standard textbooks on probability and statistics (e.g., Hamilton, 1964 Hoel et al., 1971 Lloyd, 1980 Papoulis, 1984 Dudewicz and Mishra, 1988). For most practical purposes, the statistical analysis of geochemical parameters will be restricted to the field of continuous random variables. 

Since distributions describing a discrete random variable may be less familiar than those routinely used for describing a continuous random variable, a presentation of basic theory is warranted. Count data, expressed as the number of occurrences during a specified time interval, often can be characterized by a discrete probability distribution known as the Poisson distribution, named after Simeon-Denis Poisson who first published it in 1838. For a Poisson-distributed random variable, Y, with mean X, the probability of exactly y events, for y = 0,1, 2,..., is given by Eq. (27.1). Representative Poisson distributions are presented for A = 1, 3, and 9 in Figure 27.3. 

In probability theory, the probability density function is another important fimctional quantity. In theory for a continuous random variable this is related to the distribution function by the equation ... 

Probability provides a classical model for dealing with uncertainty (Halpem 2003). The basic elements of probability theory are a) random variables and b) events, which are appropriate subsets of the sample space Q. A probabilistic model is an encoding of probabilistic information that allows the probability of events to be computed, according to the axioms of probability. In the continuous case, the usual method for specifying a probabilistic model assumes a full joint PDF over the considered random variables. 

Summation in Eq. (8.80) can be used in case where the PSD is not a continuous function but concentrates in more or less frequencies, as shown in Fig. 8.44. The determination of the moments Mj is fundamental in the fatigue analysis since it makes it possible to calculate an empirical closed-form expression for the pdf probability density function of a random spectrum variable X(t) [35], the number of zero up crossing [0] and peak [P] per second in the sample period T and the irregularity factor y given by Eq. (8.61). In theory, all the possible moments could be calculated, however, in practice, M , A/i, M2 and M4 are sufficient to calculate all of the information for the fatigue analysis. This information is ... 

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