Chapter 4 Multivariate Random Variables, Correlation, and ...

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Chapter 4Multivariate Random Variables, Correlation, and Error PropagationOf course I would not propagate error for its own be not merelywicked, but diabolical. Thomas Babington Macaulay, speechtoParliament, April 14, IntroductionSo far we have dealt with a single Random variableX,whichwecan use tomodel collections of scalar,and similar data, suchasthe distance between points orthe times between magnetic we mean by similar isthat all thedata are (we assume in the model) of the same kind of thing,ormeasurement, sothat a single,scalar serve as the probabilistic this Chapter ,wegeneralize to this to pairs,triples,...m-tuples of use suchmultiple variables either to represent vector-valued, butstill similar,quantities (suchasvelocity,the magnetic field, or angular motionsbetween tectonic plates); or we mayuse them to model situations in whichwehavetwo or more different kinds of quantities that we wish to model ,wewant to have probability models that can include cases in whichthedata appear to depend on example of sucharelationshipoccurs,for example,ifone data type is the mag

Version 1.4 Multivariate Probability 4-3 Figure 4.2 X2 falling in a certain range is not unrelated to the probability ofX1 falling in a cer- tain (perhaps different) range: for example,if X1 is around zero, X2 will tend to be; if X1 is far from zero, X2 will be positive.Wewill see how to formalize this later.Itis this ability to express relationships that makes multivariate probability suchause-

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