A Probability Distribution and Its Uses in Fitting Data

A PROBABILITY DISTRIBUTION AND ITS USES IN FITTING DATA2030.53 =0; Ca4=3,5,90.4> 0.3zLIJ0.20.1 -0.0 I I I I-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0XFIGURE 2a. Density plots for specified a, and a, values (a, = 0; a4 = 3, 5, 9).4.0ponential distribution. The non-truncated densitywas obtained using the parameter values given inTable 4; the other density demonstrates that positivelyskewed, J-shaped curves result when XA = 0.The latter curve provides a good approximation tothe exponential density. Indeed, Schmeiser [13] hasshown that the limiting distribution of this distributionis exponential with parameter 0 as X4 -- 0 when X,= XA = 0 and A2 = 4/0.The distribution can also provide good approximationsto other well-known densities. For example, thedistribution with X, = 0, X2 = 0.1975, and XA = A4 =0.1349 results in an approximation to the normaldistribution for which max, I ?(x) - R-(x)l .001,where I(x) is the normal distribution function.Although distributions are not necessarily deter-mined by their moments, the moments often do provideuseful information. In Figure 3 some distributionsare characterized by their skewness andkurtosis. The normal, the rectangular, and the exponentialdistribution are each represented by asingle point. The Student's t, the lognormal, thegamma, and the Weibull distributions are each representedby a line. The beta distribution is representedby a region of values. The proposed distribution coversthe screened area; refer to Table 4 for some of thevalues included.The Pearson and Johnson systems also cover largeregions of this diagram (see, for example, Hahn andShapiro [6]). Both of these systems incorporate anumber of functional forms whereas the proposeddistribution uses only one function and is computa-0.5 -0.4 -CQ3=1; O4=4,6,90.3 -zw0 r -U.c0.1-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0XFIGURE 2b. Density plots for specified as and a4 values (a(, = 1: (4y = 4, 6. 9).TECHNOMETRICS ?, VOL. 21, NO. 2, MAY 1979