Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sharp bounds for cumulative distribution functions   Ratios of integrals can be bounded in terms of ratios of integrands under
certain monotonicity conditions. This result, related with L'H\^{o}pital's
monotone rule, can be used to obtain sharp bounds for cumulative distribution
functions. We consider the case of noncentral cumulative gamma and beta
distributions. Three different types of sharp bounds for the noncentral gamma
distributions (also called Marcum functions) are obtained in terms of modified
Bessel functions and one additional type of function: a second modified Bessel
function, two error functions or one incomplete gamma function. For the
noncentral beta case the bounds are expressed in terms of Kummer functions and
one additional Kummer function or an incomplete beta function. These bounds
improve previous results with respect to their range of application and/or its
sharpness.

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The categories of this paper are: alg-geom
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The categories of this paper are: q-bio
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The categories of this paper are: quant-ph
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The categories of this paper are: physics
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The categories of this paper are: funct-an
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The categories of this paper are: eess
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The categories of this paper are: acc-phys
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The categories of this paper are: patt-sol
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The categories of this paper are: q-alg
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The categories of this paper are: chao-dyn
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The categories of this paper are: chem-ph
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The categories of this paper are: q-fin
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The categories of this paper are: solv-int
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The categories of this paper are: econ
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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-ex