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Interpolated approximations and bounds are all of the form ~ () + (~ ()) where ~ is an interpolating function running monotonially from 0 at low α to 1 at high α, approximating an ideal, or exact, interpolator (): = () () For the simplest interpolating function considered, a first-order rational function ~ = + the tightest lower bound has ...
File:Gamma_distribution_pdf.png licensed with Cc-by-sa-3.0-migrated, GFDL, GPL 2005-03-10T20:34:05Z MarkSweep 1300x975 (162472 Bytes) new version of PDF and matching CDF; 2005-03-10T17:45:44Z Cburnett 960x720 (138413 Bytes) Probability density function for the Gamma distribution {{GFDL}} Uploaded with derivativeFX
In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to the gamma distribution.
Repeated application of the recurrence relation for the lower incomplete gamma function leads to the power series expansion: [2] (,) = = (+) (+) = = (+ +). Given the rapid growth in absolute value of Γ(z + k) when k → ∞, and the fact that the reciprocal of Γ(z) is an entire function, the coefficients in the rightmost sum are well-defined, and locally the sum converges uniformly for all ...
A detailed analysis of this spectrum is typically used to determine the identity and quantity of gamma emitters present in a gamma source, and is a vital tool in radiometric assay. The gamma spectrum is characteristic of the gamma-emitting nuclides contained in the source, just like in an optical spectrometer , the optical spectrum is ...
The quantile function can be found by noting that (;,,) = ((/)) where is the cumulative distribution function of the gamma distribution with parameters = / and =. The quantile function is then given by inverting F {\displaystyle F} using known relations about inverse of composite functions , yielding:
One of the advantage of defining this type incomplete-version of Bessel function (,) is that even for example the associated Anger–Weber function defined in Digital Library of Mathematical Functions [6] can related:
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