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Unformatted text preview: MS&E 352 Handout #13 Decision Analysis II February 10 th , 2009 ______________________________________________________________________________ ____________________________________________________________________________ 02/11/08 page 1 of 6 Moment Matching Discretizing Probability Distributions to Two or Three Points by Matching Moments by Bill Poland, 1/96 Introduction Practical decision analysis commonly requires approximation of a continuous or many-valued probability distribution by a distribution with just a few points. This discretization, or re- discretization if the initial distribution is already discrete, simplifies probability calculations and allows conditional distributions to be assessed more easily. One approach to finding such an approximation is to match as many as possible of the first statistical moments of the initial distribution. Raw moments (moments about zero), such as the mean, or central moments, such as the variance, or a combination of these can be used. Other approaches to discretization include the equal areas method [McNamee and Celona 1987], which preserves the mean but underestimates the magnitude of higher moments. A discrete distribution with n points has 2n- 1 independent parameters: n values and n- 1 probabilities, since the last probability is such that the probabilities sum to 1. Therefore, an n- point distribution can match 2n- 1 moments of a given distribution. For example, a 2-point distribution can match the mean, variance and skewness of a given distribution. The three unknowns are the two values (say x 1 and x 2 ) and a probability (say p 1 ) of the discrete distribution, and the three equations are p 1 x 1 + p 2 x 2 = mean p 1 (x 1- mean) 2 + p 2 (x 2- mean) 2 = variance p 1 (x 1- mean) 3 + p 2 (x 2- mean) 3 = skewness where the right-hand sides are calculated from the given distribution. Calculating an n-point distribution requires solving 2n- 1 simultaneous nonlinear equations for the values and probabilities. These equations are not straightforward to solve for arbitrary n. At least two methods are available, described in Smith [1990, 1993] and Miller and Rice . MS&E 352 Handout #13 Decision Analysis II February 10 th , 2009 ______________________________________________________________________________ ____________________________________________________________________________ 02/11/08 page 2 of 6 Moment Matching Because of the programming and computational burden of calculating moment-matching discretizations for an arbitrary number of points, this paper gives formulas for two-point and...
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This note was uploaded on 06/16/2010 for the course MS&E 352 taught by Professor Ronhoward during the Winter '09 term at Stanford.
- Winter '09