A uniform distribution, sometimes also known as a rectangular distribution, is a distribution that has constant probability.
The probability density function and cumulative distribution function for a continuous uniform distribution on the interval are
(1)
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(2)
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These can be written in terms of the Heaviside step function as
(3)
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(4)
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the latter of which simplifies to the expected for .
The continuous distribution is implemented as UniformDistribution[a, b].
For a continuous uniform distribution, the characteristic function is
(5)
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If and , the characteristic function simplifies to
(6)
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(7)
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The moment-generating function is
(8)
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(9)
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(10)
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and
(11)
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The moment-generating function is not differentiable at zero, but the moments can be calculated by differentiating and then taking . The raw moments are given analytically by
(13)
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The first few are therefore given explicitly by
(16)
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The central moments are given analytically by
(20)
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(21)
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(22)
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The first few are therefore given explicitly by
(23)
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(24)
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(25)
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(26)
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The mean, variance, skewness, and kurtosis excess are therefore
(27)
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(28)
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(29)
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(30)
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