Approximating the Distribution of a Sum of Log-normal Random Variables Barry R. Cobb Virginia Military Institute Lexington, VA, USA cobbbr@vmi.edu Rafael Rum´ ı AntonioSalmer´on Universidad de Almer´ ıa, Spain rrumi,antonio.salmeron@ual.es Abstract This paper introduces a process for estimating the distribution of a sum of independent and identically distributed log-normal random variables (RVs). The procedure involves using the Fenton-Wilkinson method to estimate the parameters for a single log-normal distribution that approximates the sum of log-normal RVs. Once these parameters are determined, a mixture of truncated exponentials (MTE) function is determined to approx- imate this distribution. The MTE parameters are stated as polynomial functions of the log-normal scale parameter. Applications to inventory management are presented that demonstrate the usefulness of the MTE approximation. 1 Introduction Finding the probability density function (PDF) for a sum of log-normally distributed random variables (RVs) is an important problem in busi- ness and telecommunications (Beaulieu et al., 1995). This paper proposes a tractable approx- imation to the PDF for a sum of log-normal RVs that can be utilized in Bayesian networks (BNs) and influence diagrams (IDs). Consider the following business application in inventory management. Suppose the indepen- dent and identically distributed (i.i.d.) values of customer demand for a business, X ℓ , in each pe- riod ℓ =1,...,L, are log-normally distributed, i.e. X ℓ ∼ LN (μ X ℓ ,σ 2 X ℓ ). This means that to- tal customer demand over the L periods, X , is determined as the following sum of i.i.d. RVs: X = X 1 + X 2 + X 3 + ... + X L . (1) The value L represents the fixed lead time be- tween the placement and arrival of an inventory order. The business needs to establish order quantity and reorder point policies that mini- mize inventory costs. Finding a tractable ap- proximation to the distribution for X is impor- tant to solving this problem. Fenton (1960) and Schwartz and Yeh (1982) estimate the PDF for a sum of log-normal RVs using another log-normal PDF with the same mean and variance. The Fenton approximation (sometimes referred to as the Fenton-Wilkinson (FW) method) is simpler to apply, and for a wide range of log-normal parameters has been shown to be reasonably accurate in comparison to the Schwartz-Yeh (SY) method (Beaulieu et al., 1995). While the FW method produces a PDF that in some cases is a good approximation to the PDF for a sum of log-normals, this still does not result in a tractable representation that can be incorporated in BNs and IDs. This is be- cause such models require a functional form that can be combined with other functions, then integrated in closed-form. This paper intro- duces a new mixtures of truncated exponen- tials (MTE) approximation to the log-normal PDF. MTE functions were introduced by Moral et al. (2001) to facilitate closed-form mathe- matical calculations in BNs. One technique (Cobb et al., 2006) suggested for approximating a log-normal PDF with an MTE function requires a nonlinear optimization problem to be solved for each possible combina- tion of μ and σ 2 . In this paper, an approxi- mation is introduced that calculates the MTE parameters as polynomial functions of only the scale parameter, σ, for a range of values needed Sixth European Workshop on Probabilistic Graphical Models, Granada, Spain, 2012 67