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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