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Kotz[1] suggest the logarithmic approximation to the digamma function () ln( which results in the following approximation to the geometric mean:This makes the geometric mean the only correct mean when averaging normalized results, that is results that are presented as ratios to reference values.[14] This is relevant because the beta distribution is a suitable model for the random behavior of percentages and it is particularly suitable to the statistical modelling of proportions. The geometric mean plays a central role in maximum likelihood estimation, see section "Parameter estimation, maximum likelihood." Actually, when performing maximum likelihood estimation, besides the geometric mean GX based on the random variable X, also another geometric mean appears naturally: the geometric mean based on the linear transformation(1X), the mirror image of X, denoted by G(1X):The accompanying plots show the log geometric variances and log geometric covariance versus the shape parameters and. The plots show that the log geometric variances and log geometric covariance are close to zero for shape parameters and greater than 2, and that the log geometric variances rapidly rise in value for shape parameter values and less than unity.

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iPhone Cases