A note on the sum of uniform random variables

doi:10.1016/j.spl.2009.06.020

Statistics & Probability Letters

Volume 79, Issue 19, 1 October 2009, Pages 2092-2097

https://doi.org/10.1016/j.spl.2009.06.020 Get rights and content

Abstract

An inductive procedure is used to obtain distributions and probability densities for the sum $S_{n}$ of independent, non-equally uniform random variables. Some known results are then shown to follow immediately as special cases. Under the assumption of equally uniform random variables some new formulas are obtained for probabilities and means related to $S_{n}$ . Finally, some new recursive formulas involving distributions are derived.

Introduction

The problem of calculating the distribution of the sum $S_{n}$ of $n$ 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 $S_{n}$ 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 $S_{n}$ 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 $S_{n}$ . 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 $S_{n}$ 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 ${X_{n}}_{n \in N}$ denote a sequence of uniform distributed independent random variables and denote $S_{n} = \sum_{i = 1}^{n} X_{i}$ . Without loss of generality we assume that $X_{n} \sim U (0, a_{n})$ with $a_{n}$ positive real numbers. By adopting a suitably modified procedure due to Feller (1966) we shall obtain the probability density function $f_{n} (x)$ and the distribution function $F_{n} (x)$ of $S_{n}$ for all $n \in N$ . The starting point is to write $F_{X_{n}} (x) = \frac{x^{+} - {(x - a_{n})}^{+}}{a_{n}}, \forall n \in N, \forall x \in R,$ where ${(x - c)}^{+} = max {x - c, 0}$ , $\forall c \in R$ . Next we shall make use of $\int_{- \infty}^{x} {[{(y - c)}^{+}]}^{n - 1} d y = 1$

A special case

Let us assume that the random variables in ${X_{n}}_{n \in N}$ are identically distributed.

Proposition 3.1

When $a_{n} = a > 0$ for all $n \in N$ then $F_{n} (x) = \frac{1}{n! a^{n}} \sum_{ν = 0}^{n} {(- 1)}^{ν} (\binom{n}{ν}) {[{(x - ν a)}^{+}]}^{n}, \forall n \in N, \forall x \in R$ and $f_{n + 1} (x) = \frac{1}{n! a^{n + 1}} \sum_{ν = 0}^{n + 1} {(- 1)}^{ν} (\binom{n + 1}{ν}) {[{(x - ν a)}^{+}]}^{n}, \forall n \in N, \forall x \in R .$

Proof

Eq. (11) follows from Eq. (8) after noting that now $A_{n} = a^{n}$ and that $a_{j_{1}} + a_{j_{2}} + \dots + a_{j_{ν}} = ν a$ for $ν = 0, 1, \dots, n$ . Indeed, in the sum on $ν$ in Eq. (8), the term in curly bracket becomes ${[{(x - ν a)}^{+}]}^{n}$ , so that $\sum_{j_{1} = 1}^{n} \sum_{j_{2} = j_{1} + 1}^{n} \dots \sum_{j_{ν} = j_{ν - 1} + 1}^{n} {{[x - (a_{j_{1}} + a_{j_{2}} + \dots + a_{j_{ν}})]}^{+}}^{n} = (\binom{n}{ν}) \cdot {[{(x - ν a)}^{+}]}^{n} .$ Eq. (12) follows from (9) by a

Acknowledgements

We wish to thank Professors R. Johnson and L.M. Ricciardi for helpful comments.

References (6)

D.M. Bradley et al.
On the distribution of the sum of $n$ non-identically distributed uniform random variables
Ann. Inst. Statist. Math.
(2002)
W. Feller
An Introduction to Probability Theory and its Applications. Vol. II
(1966)
G.H. Hardy et al.
Inequalities
(1978)

There are more references available in the full text version of this article.

Cited by (0)

View full text

Statistics & Probability Letters

A note on the sum of uniform random variables

Abstract

Introduction

Section snippets

The general case

A special case

Acknowledgements

On the distribution of the sum of n non-identically distributed uniform random variables

Ann. Inst. Statist. Math.

An Introduction to Probability Theory and its Applications. Vol. II

Inequalities

On the distribution of the sum of $n$ non-identically distributed uniform random variables