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Kumaraswamy distribution

Where do you meet this distribution?

  • Hydrology

    Fletcher, S.G., and Ponnambalam, K. (1996). “Estimation of reservoir yield and storage distribution using moments analysis”. Journal of Hydrology 182: 259-275.

Shape of Distribution

Basic Properties

  • Two parameters a, b are required.
  • Continuous distribution defined on bounded range 0\leq x \leq 1
  • This distribution can be symmetric or asymmetric.

Probability

  • Cumulative distribution function
    F(x)=1-\left(1-x^a\right)^b
  • Probability density function
    f(x)=abx^{a-1}\left(1-x^a\right)^{b-1}
  • 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)
    =1-(1-A2^A3)^A4 Cumulative distribution function for the terms above
    =A3*A4*A2^(A2-1)*(1-A2^A3)^(A4-1) Probability density function for the terms above

Quantile

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

Characteristics

Mean – Where is the “center” of the distribution? (Definition)

  • Mean
    bB\left(1+\frac{1}{a},b\right)
  • How to compute this on Excel
     
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    A B
    Data Description
    8 Value of parameter A
    2 Value of parameter B
    Formula Description (Result)
    =A3*EXP(GAMMALN(1+1/A2)+GAMMALN(A3)-GAMMALN(1+1/A2+A3)) Mean of the distribution for the terms above

Random Numbers

  • Random number x is generated by inverse function method, which is for uniform random U,
    x=\left[1-\left(1-U\right)^{1/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
    0.5 Value of parameter B
    Formula Description (Result)
    =POWER(1-(1-NTRAND(100))^(1/A3),1/A2) 100 Kumaraswamy 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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