Review of probability
This chapter reviews probability theory, limiting the discussion to the study of random events as opposed to random processes, the latter being a sequence of random events extended over a period of time. The goal is to raise the level of approach by demonstrating the usefulness of delta functions. The chapter presents a calculation of the chi-squared distribution (important in statistical decision making) with delta functions. The normalisation condition of the probability density in chi-square leads to a geometric result; namely, the volume of a sphere in n dimensions can be determined without ever transferring to spherical coordinates. This chapter also discusses the first and second laws of gambling, along with distribution functions, stochastic variables, expectation values for single random variables, characteristic functions and generating functions, measures of dispersion, joint events, conditional probabilities and Bayes' theorem, sums of random variables, fitting of experimental observations, and multivariate normal distributions.
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