A note on the sum of uniform random variables
Introduction
The problem of calculating the distribution of the sum of uniform random variables has been the object of considerable attention even in recent times. The motivation can be ascribed to various reasons such as the necessity of handling data drawn from measurements characterized by different levels of precision (Bradley and Gupta, 2002), or questions appearing in change point analysis (Sadooghi-Alvandi et al., 2009), or, more in general, the need of aggregating scaled values with differing numbers of significant figures (Potuschak and Müller, 2009). It appears that this problem has been taken up first in Olds (1952), where by somewhat obscure procedures formulas for the probability density function of and its distribution function are derived. An accurate bibliography of articles published in the last century is found in Bradley and Gupta (2002), where the authors also obtain the probability density function of by non-probabilistic arguments, namely via a complicated analytical inversion of the characteristic function. Such a procedure was successively and successfully simplified in Potuschak and Müller (2009), where again no trace of probabilistic arguments is present. An attempt to achieve the same results by a simpler procedure appears in Sadooghi-Alvandi et al. (2009) where a given function is assumed to be the unknown probability density function, the proof of the correctness of such an ansatz being that its Laplace transform coincides with the moment generating function of . Quite differently, the present note includes a novel proof of the above cited results (Proposition 2.1). This is based on an inductive procedure, suitably adapted to our general instance, used by Feller (1966) for the case of identically distributed variables, that further pinpoints the usefulness of induction procedures in the probability context. (See also Hardy et al. (1978) for some more illuminating examples.) In the case of identically distributed random variables, some results concerning certain probabilities and means of random variables related to are obtained (Lemma 3.1, Theorem 3.1, Corollary 3.1, Corollary 3.2, Proposition 3.4), as well as certain recurrence relations that are reminiscent of those holding for Stirling numbers (Proposition 3.5, Proposition 3.6, Proposition 3.7).
Section snippets
The general case
Let denote a sequence of uniform distributed independent random variables and denote . Without loss of generality we assume that with positive real numbers. By adopting a suitably modified procedure due to Feller (1966) we shall obtain the probability density function and the distribution function of for all . The starting point is to write where , . Next we shall make use of
A special case
Let us assume that the random variables in are identically distributed.
Proposition 3.1 When for all thenand
Proof Eq. (11) follows from Eq. (8) after noting that now and that for . Indeed, in the sum on in Eq. (8), the term in curly bracket becomes , so that Eq. (12) follows from (9) by a
Acknowledgements
We wish to thank Professors R. Johnson and L.M. Ricciardi for helpful comments.
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