In Chapter I we learned how to handle transformations in order to find the distribution of new (constructed) random variables. Since the arithmetic mean or average of a set of (independent) random variables is a very important object in probability theory as well as in statistics, we focus in this chapter on sums of independent random variables (from which one easily finds corresponding results for the average). We know from earlier work that the convolution formula may be used but also that the sums or integrals involved may be difficult or even impossible to compute. In particular, this is the case if the number of summands is “large.” In that case, however, the central limit theorem is (frequently) applicable. This result will be proved in the chapter on convergence; see Theorem VI.5.2.
KeywordsCharacteristic Function Independent Random Variable Uniqueness Theorem Moment Generate Function Probability Generate Function
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