Search for gamma-ray burst classes with the RHESSI satellite

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Table 5:

Parameters of the best fit with two and three bivariate log-normal functions done by the maximum likelihood method on the RHESSI data. $\mu _{{x}}$ are the means on the x-axis ( $x = \log T_{90}$ ), $\mu _{{y}}$ are the means on the y-axis ( $y = \log H$ ), $\sigma _{{x}}$ are the dispersions on the x-axis, $\sigma _{{y}}$ are the dispersions on the y-axis, r are the correlation coefficients, w are the weights of the distribution and $L_{{\rm 2}}, L_{{\rm 3}}$ are the likelihoods. Given uncertainties are standard deviations of the parameters obtained by ten different fittings of data sets, where the durations and hardness ratios were randomly changed by their uncertainties.
Parameter	2 log-normal	3 log-normal
$\mu_{{x,\rm short}}$	-0.38 $\pm$ 0.018	-0.65 $\pm$ 0.018
$\mu_{{y,\rm short}}$	0.26 $\pm$ 0.010	0.26 $\pm$ 0.009
$\sigma_{{x,\rm short}}$	0.42 $\pm$ 0.012	0.20 $\pm$ 0.016
$\sigma_{{y,\rm short}}$	0.15 $\pm$ 0.009	0.15 $\pm$ 0.012
$w_{{\rm short}}[\%]$	14.2 $\pm$ 0.3	9.2 $\pm$ 0.5
$r_{{\rm short}}$	0.14 $\pm$ 0.092	0.22 $\pm$ 0.119
$\mu_{{x,\rm long}}$	1.25 $\pm$ 0.004	1.25 $\pm$ 0.006
$\mu_{{y,\rm long}}$	-0.05 $\pm$ 0.004	-0.05 $\pm$ 0.004
$\sigma_{{x,\rm long}}$	0.42 $\pm$ 0.004	0.42 $\pm$ 0.005
$\sigma_{{y,\rm long}}$	0.22 $\pm$ 0.003	0.22 $\pm$ 0.003
$w_{{\rm long}}[\%]$	85.8 $\pm$ 0.3	85.5 $\pm$ 0.8
$r_{{\rm long}}$	-0.14 $\pm~$ 0.018	-0.13 $\pm$ 0.020
$\mu_{{x,\rm middle}}$		0.11 $\pm$ 0.029
$\mu_{{y,\rm middle}}$		0.27 $\pm$ 0.019
$\sigma_{{x,\rm middle}}$		0.21 $\pm$ 0.057
$\sigma_{{y,\rm middle}}$		0.17 $\pm$ 0.035
$w_{{\rm middle}}[\%]$		5.3 $\pm$ 1.1
$r_{{\rm middle}}$		0.59 $\pm$ 0.230
$\ln L_{{\rm 2}}$	-323.91
$\ln L_{{\rm 3}}$		-313.00

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