Difference between revisions of "GammaIInv"

 
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[[category:Analytic Distribution Functions]]
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[[Category:Analytic Distribution Functions]]
[[Category:Doc Status D]] <!-- For Lumina use, do not change -->
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[[Category: Distribution Densities library functions]]
  
= GammaIInv(Y, A, B) =
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== GammaIInv(y, a, b) ==
  
Returns the incomplete gamma function, defined as:
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Returns the inverse of the incomplete [[Gamma]] function. The function returns the value ''x'' such that
:<math>GammaI(X,A,B) = {1\over{\gamma(A)}} \int_0^{x/B} e^{-t} t^{A-1} dt</math>
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:[[GammaI]](x, a, b) = y
  
''A'' is the shape parameter, ''B'' is an optional scale factor (default ''B=1'').  Note that some textbooks use <math>\lambda = 1/A</math>as the scale factor.  The incomplete gamma function is defined for <math>X\geq 0</math>.
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«a» is the shape parameter, «b» is an optional scale factor that defaults to 1.
  
The incomplete gamma function returns the cumulative area from zero to ''X'' under the [[Gamma]] distribution.
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[[GammaIInv]]([[Uniform]](0, 1), a, b) is equivalent to [[Gamma]](a, b).
  
The incomplete gamma function is useful in a number of mathematical and statistical contexts.
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[[GetFract]]([[Gamma]](a, b), y) is an approximation to [[GammaIInv]](y, a, b), obtained via sampling. The former is subject to sampling error, with less error at increased sample sizes, whereas by using [[GammaIInv]] directly, sampling error is avoided entirely.
  
= Library =
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== Library ==
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Advanced Math, [[Distribution Densities Library]]  ([[media:Distribution Densities.ana|Distribution Densities.ana]])
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:Use [[File menu|File]] &rarr; '''Add Library...''' to add this library
  
Advanced Math
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== Examples ==
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:<code>Index y := Sequence(0,0.98, 0.01)</code>
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:<code>Variable a := [0.5, 1, 2, 5]</code>
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:<code>GammaIInv(y, a) &rarr;</code>
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:[[image:GammaIInv.png]]
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:<code>GammaI(12, 5, 2) &rarr; 0.7149 </code>
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:<code>GammaIInv(0.7149, 5, 2) &rarr; 12 </code>
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==See Also==
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* [[Gamma]]
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* [[GammaI]]
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* [[Distribution Densities Library]]
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* [[media:Distribution Densities.ana|Distribution Densities.ana]]

Latest revision as of 21:16, 24 May 2016


GammaIInv(y, a, b)

Returns the inverse of the incomplete Gamma function. The function returns the value x such that

GammaI(x, a, b) = y

«a» is the shape parameter, «b» is an optional scale factor that defaults to 1.

GammaIInv(Uniform(0, 1), a, b) is equivalent to Gamma(a, b).

GetFract(Gamma(a, b), y) is an approximation to GammaIInv(y, a, b), obtained via sampling. The former is subject to sampling error, with less error at increased sample sizes, whereas by using GammaIInv directly, sampling error is avoided entirely.

Library

Advanced Math, Distribution Densities Library (Distribution Densities.ana)

Use FileAdd Library... to add this library

Examples

Index y := Sequence(0,0.98, 0.01)
Variable a := [0.5, 1, 2, 5]
GammaIInv(y, a) →
GammaIInv.png
GammaI(12, 5, 2) → 0.7149
GammaIInv(0.7149, 5, 2) → 12

See Also

Comments


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