Suppose a sample of size n of vectors ( X _ i, Y _ i )'is taken from a bivariate normal distribution with unknown correlation \ rho.
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Under the null hypothesis of that \ rho = \ rho _ 0, given the assumption that the sample pairs are independent and identically distributed and follow a bivariate normal distribution.
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Note that the bivariate normal distribution is defined over the entire plane, while the mean is confined to be in the unit ball ( on or inside the unit circle ).
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The Fisher transformation is an approximate variance-stabilizing transformation for " r " when " X " and " Y " follow a bivariate normal distribution.
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It can be shown that the distribution of [ \ overline { C }, \ overline { S } ] approaches a bivariate normal distribution in the limit of large sample size.
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For a bivariate normal distribution, the cokurtosis tensor has off-diagonal terms that are neither 0 nor 3 in general, so attempting to " correct " for an excess becomes confusing.
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If the local distribution function is a parametric distribution, such as a symmetric bivariate normal distribution then the method is referred to as a "'kernel method "', but more correctly should be designated as a parametric kernel method.
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MSWD = 1 if the age data fit a univariate normal distribution in t ( for the arithmetic mean age ) or log ( " t " ) ( for the geometric mean age ) space, or if the compositional data fit a bivariate normal distribution in [ log ( U / He ), log ( Th / He ) ]-space ( for the central age ).
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