The Kumaraswamy distribution, also known as the double bounded distribution, is a continuous probability distribution defined on the interval [0, 1]. It's named after Ponnuthurai Kumaraswamy.
Key Features:
Bounded Support: The random variable always falls between 0 and 1, making it suitable for modeling proportions, fractions, or percentages.
Flexibility: The distribution's shape is determined by two shape parameters, a and b, which allow it to take on a wide range of forms, including shapes similar to the beta distribution.
Applications: It is used in various fields like hydrology, economics, and machine learning to model quantities bounded between 0 and 1. It has been used as an alternative to the beta distribution due to its simpler closed-form expressions for distribution and quantile functions.
Probability Density Function (PDF):
The PDF of the Kumaraswamy distribution is given by:
f(x; a, b) = a * b * x^(a-1) * (1 - x^a)^(b-1) for 0 ≤ x ≤ 1, a > 0, b > 0
where a and b are shape parameters.
Cumulative Distribution Function (CDF):
The CDF is given by:
F(x; a, b) = 1 - (1 - x^a)^b for 0 ≤ x ≤ 1, a > 0, b > 0
Applications in Hydrology: The Kumaraswamy distribution has found particular utility in modeling hydrological variables, such as rainfall intensities or streamflow fractions, which are often expressed as proportions or percentages. The distribution's flexibility in capturing various distributional shapes makes it a useful tool for water resources management.
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