The probability integral transform states that if X is a continuous random variable with cumulative distribution function F _ X, then the random variable Y = F _ X ( X ) has a uniform distribution on [ 0, 1 ].
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In mathematics, the "'Kontorovich Lebedev transform "'is an integral transform which uses a Macdonald function ( modified Bessel function of the second kind ) with imaginary index as its "'kernel " '.
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The concept underlying the method is based on the probability integral transform, in that a set of independent random samples derived from any random variable should on average be uniformly distributed with respect to the cumulative distribution function of the random variable.
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The integral transform part is some integral transform of " g " and the spectral part is a sum of Fourier coefficients, taken over spaces of holomorphic and non-holomorphic modular forms twisted with some integral transform of " g ".
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The integral transform part is some integral transform of " g " and the spectral part is a sum of Fourier coefficients, taken over spaces of holomorphic and non-holomorphic modular forms twisted with some integral transform of " g ".
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The integral transform part is some integral transform of " g " and the spectral part is a sum of Fourier coefficients, taken over spaces of holomorphic and non-holomorphic modular forms twisted with some integral transform of " g ".
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:: More precisely, see for example pages 256-8, Distributions in the Physical and Engineering Sciences : Volume 1 : Distributional and Fractal Calculus, Integral Transforms and Wave, ISBN : 0817639241 Twma 02 : 46, 3 April 2006 ( UTC)
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For the special case that the non-zero extent of both " x " and " h " are " d " N ", this is reducible to matrix multiplication where the kernel of the integral transform is a circulant matrix.
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If both the probability density and its logarithm can be expressed as a Fourier series ( or more generally, any integral transform on the circle ) then the orthogonality property may be used to obtain a series representation for the entropy which may reduce to a closed form.
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The Integral transform article says that any basis functions of an integral transform have to be orthogonal, and then says that the basis functions of the Fourier transform are \ frac { e ^ {-iut } } { \ sqrt { 2 \ pi } }.
