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Gumbel (Type II) distribution

Where do you meet this distribution?

Shape of Distribution

Basic Properties

  • Two parameters a, b are required.
    a>0, b>0
  • Continuous distribution defined on semi-infinite range x>0
  • This distribution is always asymmetric.

Probability

  • Cumulative distribution function
    F(x)=\exp\left(-bx^{-a}\right)
  • Probability density function
    f(x)=abx^{-a-1}\exp\left(-bx^{-a}\right)
  • How to compute these on Excel.
     
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    A B
    Data Description
    0.5 Value for which you want the distribution
    8 Value of parameter A
    2 Value of parameter B
    Formula Description (Result)
    =EXP(-A4*A2^(-A3)) Cumulative distribution function for the terms above
    =A3*A4*A2^(-A3-1)*EXP(-A4*A2^(-A3)) Probability density function for the terms above

Quantile

  • Inverse function of cumulative distribution function
    F^{-1}(P)=\left(-\frac{\ln P}{b}\right)^{-1/a}
  • How to compute this on Excel.
     
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    A B
    Data Description
    0.7 Probability associated with the distribution
    1.7 Value of parameter A
    0.9 Value of parameter B
    Formula Description (Result)
    =POWER(-LN(A2)/A4,-1/A3) Inverse of the cumulative distribution function for the terms above

Random Numbers

  • Random number x is generated by inverse function method, which is for uniform random U,
    x=\left(-\frac{\ln U}{b}\right)^{-1/a}
  • How to generate random numbers on Excel.
     
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    A B
    Data Description
    0.5 Value of parameter A
    2 Value of parameter B
    Formula Description (Result)
    =POWER(-A2*LN(-NTRAND(100)/A3),1/A2) 100 Gumbel (Type II) deviates based on Mersenne-Twister algorithm for which the parameters above

    Note The formula in the example must be entered as an array formula. After copying the example to a blank worksheet, select the range A5:A104 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

NtRand Functions

Not supported yet

Reference

 

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