# 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 are required.
- Continuous distribution defined on bounded range
- This distribution can be symmetric or asymmetric.

### Probability

- Cumulative distribution function
- Probability density function
- How to compute these on Excel.

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7

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
- 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
- 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,
- 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