The concept of cumulative probability provides the basis for discussion of probabilities in finite-dimensional spaces. Most probabilities in physics are either discrete or absolutely continuous or a mixture of the two, but the notion of singular continuous distributions is needed to complete the conceptual framework. The main applications are to quantum mechanics, discussed in the next chapter, statistical mechanics, error analysis, and Monte Carlo. The outstanding phenomenon is the trend toward the universal so-called normal distributions on averaging, described in the central limit theorem. Cumulative, marginal, and conditional probabilities are used in Monte Carlo simulation by computer of natural random phenomena too complicated for analysis. The ideas of probability and measure as set functions are needed for discussion of probability in infinite-dimensional and abstract spaces, such as appear in statistical mechanics and in the theory of stochastic processes.
KeywordsUnivariate and multivariate probability distributions cumulative probabilities densities canonical decomposition of a nondecreasing function discrete, atomic, singular, continuous, and absolutely continuous probability distributions nondecreasing functions of several variables mean: expectation moments standard deviation characteristic function correlation coefficient and matrix measures set functions the extension and Riesz representation theorems for measures sampling sample mean sample variance marginal and conditional probabilities normal distribution central limit theorem the Monte Carlo method probability and measure in a Hilbert space
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