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Parameters a > 0 {displaystyle a>0,} (real) b > 0 {displaystyle b>0,} (real) Support x ∈ [ 0 , 1 ] {displaystyle xin [0,1],} PDF a b x a − 1 ( 1 − x a ) b − 1 {displaystyle abx^{a-1}(1-x^{a})^{b-1},} CDF [ 1 − ( 1 − x a ) b ] {displaystyle [1-(1-x^{a})^{b}],} Mean b Γ ( 1 + 1 a ) Γ ( b ) Γ ( 1 + 1 a + b ) {displaystyle {rac {bGamma (1+{ frac {1}{a}})Gamma (b)}{Gamma (1+{ frac {1}{a}}+b)}},} Median ( 1 − 2 − 1 / b ) 1 / a {displaystyle left(1-2^{-1/b}ight)^{1/a}} |
In probability and statistics, the Kumaraswamy's double bounded distribution is a family of continuous probability distributions defined on the interval [0,1]. 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.
Contents
Probability density function
The probability density function of the Kumaraswamy distribution is
and where a and b are non-negative shape parameters.
Cumulative distribution function
The cumulative distribution function is
Generalizing to arbitrary interval support
In its simplest form, the distribution has a support of [0,1]. In a more general form, the normalized variable x is replaced with the unshifted and unscaled variable z where:
Properties
The raw moments of the Kumaraswamy distribution are given by:
where B is the Beta function. The variance, skewness, and excess kurtosis can be calculated from these raw moments. For example, the variance is:
The Shannon entropy (in nats) of the distribution is:
where
Relation to the Beta distribution
The Kumaraswamy distribution is closely related to Beta distribution. Assume that Xa,b is a Kumaraswamy distributed random variable with parameters a and b. Then Xa,b is the a-th root of a suitably defined Beta distributed random variable. More formally, Let Y1,b denote a Beta distributed random variable with parameters
with equality in distribution.
One may introduce generalised Kumaraswamy distributions by considering random variables of the form
Note that we can reobtain the original moments setting
Related distributions
Example
A good example of the use of the Kumaraswamy distribution is the storage volume of a reservoir of capacity zmax whose upper bound is zmax and lower bound is 0 (Fletcher & Ponnambalam, 1996).