The size distributions of 20 main crystal types were approximated by single normal or log-

normal distributions; however, in most cases the combination of two normal or log

normal distributions was more suitable.

A random variable X has a two-piece

normal distribution with parameters [Mu], [[Sigma].

8 In fact, the traditional jump diffusion model can be interpreted as allowing for the possibility of a jump between an infinite number of

normal distributions (Kon [1984]).

To aid the user in making this assessment, the normal probability plot is plotted with a straight line corresponding to the

normal distribution with the mean and variance estimated from the data.

Suppose that an element is chosen at random from a

normal distribution for which the value of the mean |Theta~ is unknown (-|infinity~ |is less than~ |Theta~ |is less than~ |infinity~), and the value of the variance ||Sigma~.

In reviewing previous market events, such as the sovereign debt crisis in March 2010, the threat of potential losses was exposed with PerTrac RiskPlus, but

normal distribution analytics continued with its overly optimistic market forecast despite the abundance of warning signs.

The

normal distribution describes the distribution of randomly occurring events when nothing intervenes.

Likewise, the log-log plot of the density function for a log

normal distribution is a straight line for a large part of the body.

and replacing parameters of the

normal distribution with their definitions 20 and 21, the benchmark value is derived in the following way:

METHODS: We have presented the likelihood function for a bivariate

normal distribution taking into account values < LOD as well as missing data assumed missing at random, and we use the estimated distributional parameters to impute values < LOD and to generate multiple plausible data sets for analysis by standard statistical methods.

Figures 1-3 show examples of the frequent spectra which came from the

normal distributions being affected by two, three, and four perturbing factors (the progression coefficients [q.

Bhattacharya's method for estimating parameters for a mixture of

normal distributions is well described by Sparre & Venema (1992).