Kumaraswamy Distribution

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In probability and statistics, the **Kumaraswamy's double bounded distribution** is a family of continuous probability distributions defined on the interval differing in the values of their two non-negative shape parameters, *a* and *b*.

It is similar to the Beta distribution, but much simpler to use especially in simulation studies due to the simple closed form of both its probability density function and cumulative distribution function. This distribution was originally proposed by Poondi Kumaraswamy for variables that are lower and upper bounded.

## Characterization

### Probability density function

The probability density function of the Kumaraswamy distribution is

### Cumulative distribution function

The cumulative distribution function is therefore

### Generalizing to arbitrary range

In its simplest form, the distribution has a range of . In a more general form, the normalized variable *x* is replaced with the unshifted and unscaled variable *z* where:

The distribution is sometimes combined with a "spike probability" or a Dirac delta function, e.g.:

## Properties

The raw moments of the Kumaraswamy distribution are given by :

where*B* is the Beta function. The variance, skewness, and excess...

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It is similar to the Beta distribution, but much simpler to use especially in simulation studies due to the simple closed form of both its probability density function and cumulative distribution function. This distribution was originally proposed by Poondi Kumaraswamy for variables that are lower and upper bounded.

- <math> f(x; a,b) = a b x^^.</math>

- <math>F(x; a,b)=1-(1-x^a)^b. </math>

- <math> x = frac , qquad z_ le z le z_. ,!</math>

The distribution is sometimes combined with a "spike probability" or a Dirac delta function, e.g.:

- <math> g(x|a,b) = F_0delta(x)+(1-F_0)a b x^^.</math>

- <math>m_n = frac = bB(1+n/a,b),</math>

where

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