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Handout 4_2010

# Handout 4_2010 - 1 IE 372 Simulation Handout 4 SELECTING...

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1 IE 372 Simulation - Handout 4 SELECTING INPUT PROBABILITY DISTRIBUTIONS (Law & Kelton) For simulating a real life system, we need to estimate the probability distribution of an input random variable using past data, if any. We have two options: 1. Fit a theoretical distribution, such as exponential, normal, beta and so on. 2. Define an empirical distribution. In general, we first try to fit the data at hand to a theoretical distribution. If that fails then we define an empirical distribution. We prefer a theoretical distribution because, with an empirical distribution, the values our random variable can take (the random variates that will be generated during the simulation) are restricted by the range of the data at hand. Fitting a Theoretical Distribution There are three steps involved. 1. Choose the family of distributions, such as exponential, normal or beta. To do this: Plot the histogram of data and compare its shape with known pdf shapes. Compare point (or descriptive) statistic values, such as coefficient of variation or skewness, with known values of theoretical distributions. For example, 2 2 σ μ = for exponential or skewness = 0 for normal. 2. Estimate distribution parameters from data, e.g. mean for exponential, mean and variance for normal. 3. Test how well the data fits the distribution by using goodness-of-fit tests. That is, could this data be a sample from the hypothesized distribution? We will discuss: Chi-square goodness-of-fit test Kolmogorov-Smirnov goodness-of-fit test

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2 Some Theoretical Distributions Available in ARENA
3 Chi-Square Goodness-of-Fit Test Given an iid sample 1 , , n X X K , we want to test: H 0 : The sample 1 , , n X X K comes from the hypothesized theoretical distribution function ˆ F , against the alternative H 1 : It does not. We start by preparing the observed and expected frequency distributions. j Interval, 1 [ , ) j j a a - Observed frequency, j N Expected frequency, j np 1 0 1 [ , ) a a 1 N 1 np 2 1 2 [ , ) a a 2 N 2 np k 1 [ , ) k k a a - k N k np Total = n j p needed to calculate the expected frequencies are found as follows. 1 ˆ ( ) j j a j a p f x dx - = , 1, , j k = K , if X is continuous 1 ˆ( ) j i j j i a x a p p x - < = , 1, , j k = K , if X is discrete where ˆ f or ˆ p represents the fitted distribution. The test statistic is 2 2 1 ( ) k j j j j N np np χ = - = . Reject H 0 if 2 2 , 1 df α χ χ - where 1 df k = - if distribution parameters are known, 1 df k p = - - if p distribution parameters are estimated from the same sample. 2 , 1 df α χ - α

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4 Notes: Chi-square goodness-of-fit test can be used for both continuous and discrete distributions. Test is valid with large n ( n > 50). Choice of k : Some rules are proposed in literature, e.g. Sturges’s rule is 2 10 1 log 1 3.322log k n n = + = + In general, such rules are not very useful. You should try various k values and choose the largest k that gives a “smooth” histogram. (If k is too large then the histogram will have a “ragged” shape. If k is too small then the histogram will
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Handout 4_2010 - 1 IE 372 Simulation Handout 4 SELECTING...

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