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Getting Started

A • STATISTICAL

A • STATISTICAL DISTRIBUTIONS Beta(, ): BETA(Beta, Alpha) Probability Density Function f(x) = x – 1 1 – x – 1 --------------------------------------- B( , ) 0 otherwise where is the complete beta function given by 1 B = t – 1 1 – t – 1 dt 0 for 0 < x < 1 Parameters Shape parameters Beta () and Alpha () specified as positive real numbers. Range [0, 1] (Can also be transformed to [a,b] as described below) Mean ------------ + Variance a ---------------------------------------------- + a 2 + a + 1 Applications Because of its ability to take on a wide variety of shapes, this distribution is often used as a rough model in the absence of data. Also, because the range of the beta distribution is from 0 to 1, the sample X can be transformed to the scaled beta sample Y with the range from a to b by using the equation Y = a + (b - a)X. The beta is often used to represent random proportions, such as the proportion of defective items in a lot. 153

GETTING STARTED WITH ARENA Continuous: CONTINUOUS(CumP 1 , Val 1 , . . ., CumP n , Val n ) (c 1 , x 1 ,. . ., c n , x n ) Probability Density Function Cumulative Distribution Function c 1 if x = x 1 (a mass of probability c 1 at x 1 ) f(x) = c j – c j-1 if x j-1 x < x j , for j = 2, 3, . . ., n 0 if x < x 1 or x x n Parameters The CONTINUOUS function in Arena returns a sample from a user-defined distribution. Pairs of cumulative probabilities c j (= CumP j ) and associated values x j (= Val j ) are specified. The sample returned will be a real number between x 1 and x n , and will be less than or equal to each x j with corresponding cumulative probability c j . The x j ’s must increase with j. The c j ’s must all be between 0 and 1, must increase with j, and c n must be 1. 154

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