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Encyclopedia > Kumaraswamy distribution
Kumaraswamy
Probability density function
Probability density function
Cumulative distribution function
Probability density function
Parameters (real)
(real)
Support
pdf
cdf
Mean
Median
Mode
Variance
Skewness
Kurtosis
Entropy
mgf
Char. func.

In probability theory and statistics, Kumaraswamy's double bounded distribution is as versatile as the Beta distribution, but much simpler to use especially in simulation studies as it has a simple closed form solution for both its pdf and cdf. The distribution form was originally presented by Poondi Kumaraswamy for variables that are lower and upper bounded. In its simplest form, we can take the bounds to be in which case the probability density function is: Download high resolution version (1201x901, 87 KB) This image needs to be cleaned up to conform to a higher standard of quality. ... Download high resolution version (1201x901, 84 KB)Kumaraswamy distribution cumulative distribution function File links The following pages link to this file: Kumaraswamy distribution ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or... In probability (and especially gambling), the expected value (or (mathematical) expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds... In probability theory and statistics, the median is a number that separates the highest half of a sample, a population, or a probability distribution from the lowest half. ... In statistics, the mode is the value that has the largest number of observations, namely the most frequent value or values. ... In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ... In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ... Entropy of a Bernoulli trial as a function of success probability. ... In probability theory and statistics, the moment-generating function of a random variable X is The moment-generating function generates the moments of the probability distribution, as follows: If X has a continuous probability density function f(x) then the moment generating function is given by where is the ith... Some mathematicians use the phrase characteristic function synonymously with indicator function. The indicator function of a subset A of a set B is the function with domain B, whose value is 1 at each point in A and 0 at each point that is in B but not in A... Probability theory is the mathematical study of probability. ... Statistics is a type of data analysis whose practice includes the planning, summarizing, and interpreting of observations of a system possibly followed by predicting or forecasting of future events based on a mathematical model of the system being observed. ... In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function (pdf) defined on the interval [0, 1]: where and are parameters that must be greater than zero and is the beta function. ... In mathematics, an equation or system of equations is said to have a closed-form solution if, and only if, at least one solution can be expressed analytically in terms of a bounded number of well-known operations. ... Ponnambalam Kumaraswamy ( often referred to as Poondi Kumaraswamy) was a leading hydrologist of India with much of his work having practical significance. ...

The cumulative distribution function is therefore:

The a and b are shape parameters. In a more general form, we may replace the normalized variable x with the unshifted and unscaled variable z where:

The distribution is sometimes combined with a "pike probability" or a Dirac delta function, e.g.: The Dirac delta function, sometimes referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere such that the total integral...

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, 1996).


The raw moments of the Kumaraswamy distribution are given by: Moment refers to either of two related concepts in mathematics and physics: Moment (physics) Moment (mathematics) See also Moment (magazine), a Jewish general publication. ...

References

  • Kumaraswamy, P. (1980). "A generalized probability density function for double-bounded random processes". Journal of Hydrology, 46, 79–88.
  • Fletcher, S.G., and Ponnambalam, K. (1996). "Estimation of reservoir yield and storage distribution using moments analysis". Journal of Hydrology, 182, 259–275.

 
 

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