On computing the distribution function of the sum of independent random variables

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Abstract

We present an efficient approach to the determination of the convolution of random variables when their probability density functions are given as continuous functions over finite support. We determine that the best approach for large activity networks is to descretize the density function using Chebychev's points.

Scope and purpose

The convolution operation occurs frequently in the analysis of sums of independent random variables. Although the operation of convolution is elementary in its concept, it is rather onerous to implement on the computer for several reasons that are spelled out in the paper. It is our objective to present a computational scheme, based on discretization of continuous density functions, that is both easy to implement and mathematically “correct”, in the sense of satisfying equality of several moments of the approximation to the moments of the original density function. The approach presented in the paper can be easily programmed on the computer, and gives the desired convolution to any desired degree of accuracy.

Section snippets

The convolution of random variables

In the mathematical analysis of activity networks (ANs) one is usually faced with the problem of evaluating the sums and maxima of independent (or approximately independent) random variables (r.v's). It is well known that the maximum of two r.v's is easily secured by multiplying their probability cumulative distribution functions (c.d.f.). The sum of two independent r.v's, however, leads in a natural way to the evaluation of their convolution, which is a rather demanding operation, and

Polynomial approximation

The polynomial approximations we propose are for finite and closed domains of the mutually independent variables. We recognize that the d.f. fX(t) of the r.v. X, or simply f(t) when there is no ambiguity about the reference to the r.v., is often defined for t∈[0,∞).2 Typically, f(t) approaches zero asymptotically. For modeling purposes of almost all real-life projects we take the range to be [0,T̄], where is the upper bound on the duration

The convolution of polynomials

The proposed algorithm using polynomial approximations can be used successfully for small-size ANs. For medium to large-size networks, the computational burden places a heavy penalty on its use. This increase in the computational complexity is due to two reasons. The first is the growth of the order of the polynomial with each convolution operation. And the second is the exponential increase, O(3n−1), in the number of subintervals n over which the convolution is defined. Then one must resort to

Approximation through discretization

Discretization of a continuous d.f. is a popular method for implementing project planning algorithms. The popularity is attributed to the ease of convolution (and maximum) operations albeit at the cost of reduced accuracy. Nonetheless, the computational ease enables one to model large-scale systems.

The most common methods used to discretize a continuous density function, along with their relative pro's and con's, are the following (for a more extensive discussion of these approaches, see [6]):

Mani K. Agrawal is the Chief Scientist at Manugistics, Inc., stationed in Englewood, Colorado. He has earned his baccalaureate degree from the Indian Institute of Technology in Kanpur, and his master's degree from Vanderbilt University, 1989, both in Civil Engineering and his doctorate degree at NCSV, 1999. He has worked as consultant before joining the University of Illinois, Urbana, IL, for one semester, after which he joined Manugistics.

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Mani K. Agrawal is the Chief Scientist at Manugistics, Inc., stationed in Englewood, Colorado. He has earned his baccalaureate degree from the Indian Institute of Technology in Kanpur, and his master's degree from Vanderbilt University, 1989, both in Civil Engineering and his doctorate degree at NCSV, 1999. He has worked as consultant before joining the University of Illinois, Urbana, IL, for one semester, after which he joined Manugistics.

Salah E. Elmaghraby is University Professor of Industrial Engineering and Operations Research at NCSU. He earned his baccalaureate degree in Mechanical Engineering from the University of Cairo, his master's degree from Ohio State University and his doctorate from Cornell University, in Industrial Engineering. His areas of interest are Capacity Planning and Job Scheduling, Activity Networks, and Dynamic Programming. He is author or co-author of six books and over 86 papers in scientific journals.

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