We derive the probability density of the ratio of components of the bivariate normal distribution with arbitrary parameters. The density is a product of two factors, the first is a Cauchy density, the second a very complicated function. We show that the distribution under study does not possess an expected value or other moments of higher order. Our particular interest is focused on the shape of the density. We introduce a shape parameter and show that according to its sign the densities are classified into three main groups. As an example, we derive the distribution of the ratio Z = − Bm−1 /(mBm ) for a polynomial regression of order m. For m=1, Z is the estimator for the zero of a linear regression, for m = 2 , an estimator for the abscissa of the extreme of a quadratic regression, and for m = 3 , an estimator for the abscissa of the inflection point of a cubic regression.
Using electron microscopy, we studied the morphology of secretory granules in rat pars intermedia cells. We found figures of apparent intergranule fusion, characterized by a tight association of two granules. The fusion was detected in around 2% of all measured granules, indicating a low occurrence of intergranule fusion. To study whether intergranule fusion affects the distribution of granule diameters a simple probabilistic model was developed. It is based on the theory that larger granules are formed by fusion of two or more spherical granules of fixed size, and that the surface of a newly formed granule is equal to the sum of fused granule membranes. The model accounts for the bias on granule diameter measurements due to sectioning of granules. Although the electron microscopy data strongly indicates the existence of intergranule fusion in rat melanotrophs, this process as modelled in the present work does not contribute to the granule diameter distribution significantly. It is likely that in addition to the fusion of larger granules, other processes, such as fusion of microvesicles, may affect the distribution of granule diameters.
The classification of three-dimensional zeropotent algebras over an arbitrary field is given. It is complete up to the individual properties of the ground field.
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