Abstract
This chapter discusses the applications of independent random variables in number theory and in statistics. One of the fundamental problems of statistics is to estimate unknown parameters—for example, mean value, variance—of a distribution function F(x) from a sample where, at least in the simplest case, ξ1, ξ2,… are independent, identically distributed random variables. Each law of large numbers can be formulated as a theorem of estimation theory. The chapter presents the estimation of the distribution and density functions. The laws of large numbers imply some results on the estimations of density functions; however, the clearest results are not straightforward consequences of the laws of large numbers but concern the estimation of n-dimensional distribution functions or of probability measures defined on abstract spaces.
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