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# Getting Started

## A • STATISTICAL

A • STATISTICAL DISTRIBUTIONS Johnson: JOHNSON(Gamma, Delta, Lambda, Xi) or JOHN(Gamma, Delta, Lambda, Xi) Probability Density Function Parameters Gamma shape parameter (), Delta shape parameter (), Lambda scale parameter (), and Xi location parameter (). Range ( ) Unbounded Family [ ] Bounded Family Applications The flexibility of the Johnson distribution allows it to fit many data sets. Arena can sample from both the unbounded and bounded form of the distribution. If Delta () is passed as a positive number, the bounded form is used. If Delta is passed as a negative value, the unbounded form is used with | | as the parameter. 161

GETTING STARTED WITH ARENA Lognormal(, ): LOGNORMAL(LogMean, LogStd) or LOGN(LogMean, LogStd) Probability Density Function Denote the user-specified input parameters as LogMean = l and LogStd = l . Then let and the probability density function can then be written as f(x) = for x > 0 0 otherwise Parameters Mean LogMean ( l > 0) and standard deviation LogStd ( l > 0) of the lognormal random variable. Both LogMean and LogStd must be specified as strictly positive real numbers. Range [0, + Mean LogMean = 1 = e + 2 2 Variance (LogStd) 2 = 2 1 = e 2 + 2 e 2 – 1 Applications The lognormal distribution is used in situations in which the quantity is the product of a large number of random quantities. It is also frequently used to represent task times that have a distribution skewed to the right. This distribution is related to the normal distribution as follows. If X has a lognormal ( l , l ) distribution, then ln(X) has a normal (, ) distribution. Note that and are not the mean and standard deviation of the lognormal random variable X, but rather the mean and standard deviation of the normal random variable lnX. 162

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