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Bottom-up heapsort is a variant that reduces the number of comparisons required by a significant factor. While ordinary "top-down" heapsort requires 2n log 2 n + O(n) comparisons worst-case and on average, [10] the bottom-up variant requires n log 2 n + O(1) comparisons on average, [10] and 1.5n log 2 n + O(n) in the worst case. [11]
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables in the subset without reference to the values of the other variables.
In statistics, especially in Bayesian statistics, the kernel of a probability density function (pdf) or probability mass function (pmf) is the form of the pdf or pmf in which any factors that are not functions of any of the variables in the domain are omitted. [1] Note that such factors may well be functions of the parameters of the
[[Category:Statistics user templates]] to the <includeonly> section at the bottom of that page. Otherwise, add <noinclude>[[Category:Statistics user templates]]</noinclude> to the end of the template code, making sure it starts on the same line as the code's last character.
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Spaces within a formula must be directly managed (for example by including explicit hair or thin spaces). Variable names must be italicized explicitly, and superscripts and subscripts must use an explicit tag or template. Except for short formulas, the source of a formula typically has more markup overhead and can be difficult to read.
Excel maintains 15 figures in its numbers, but they are not always accurate; mathematically, the bottom line should be the same as the top line, in 'fp-math' the step '1 + 1/9000' leads to a rounding up as the first bit of the 14 bit tail '10111000110010' of the mantissa falling off the table when adding 1 is a '1', this up-rounding is not undone when subtracting the 1 again, since there is no ...
A variety of templates and styles are available to create timelines. The {{Graphical timeline}} template allows representations of extensive timelines. The template offers complex formatting and labeling options to control the output. Typically, each use is made into its own template, and the template is then transcluded into the article.