Mathematical Methods for Physics and Engineering - Matematica.NET

30ProbabilityAll scientists will know the importance of experiment and observation and,equally, be aware that the results of some experiments depend to a degree onchance. For example, in an experiment to measure the heights of a random sampleof people, we would not be in the least surprised if all the heights were found tobe different; but, if the experiment were repeated often enough, we would expectto find some sort of regularity in the results. Statistics, which is the subject of thenext chapter, is concerned with the analysis of real experimental data of this sort.First, however, we discuss probability. To a pure mathematician, probability is anentirely theoretical subject based on axioms. Although this axiomatic approach isimportant, and we discuss it briefly, an approach to probability more in keepingwith its eventual applications in statistics is adopted here.We first discuss the terminology required, with particular reference to theconvenient graphical representation of experimental results as Venn diagrams.The concepts of random variables and distributions of random variables are thenintroduced. It is here that the connection with statistics is made; we assert thatthe results of many experiments are random variables and that those results havesome sort of regularity, which is represented by a distribution. Precise definitionsof a random variable and a distribution are then given, as are the definingequations for some important distributions. We also derive some useful quantitiesassociated with these distributions.30.1 Venn diagramsWe call a single performance of an experiment a trial and each possible resultan outcome. Thesample space S of the experiment is then the set of all possibleoutcomes of an individual trial. For example, if we throw a six-sided die then thereare six possible outcomes that together form the sample space of the experiment.At this stage we are not concerned with how likely a particular outcome might1119