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Uniform Distribution

المؤلف:  Beyer, W. H.

المصدر:  CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press

الجزء والصفحة:  pp. 531 and 533

15-4-2021

2554

Uniform Distribution

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

		{0 for x<a; 1/(b-a) for a<=x<=b; 0 for x>b" src="https://mathworld.wolfram.com/images/equations/UniformDistribution/Inline4.gif" style="height:82px; width:124px" />	(1)
		{0 for x<a; (x-a)/(b-a) for a<=x<=b; 1 for x>b." src="https://mathworld.wolfram.com/images/equations/UniformDistribution/Inline7.gif" style="height:78px; width:124px" />	(2)

These can be written in terms of the Heaviside step function  as

			(3)
			(4)

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)

If  and , the characteristic function simplifies to

			(6)
			(7)

The moment-generating function is

			(8)
			(9)
		{(e^(tb)-e^(ta))/(t(b-a)) for t!=0; 1 for t=0," src="https://mathworld.wolfram.com/images/equations/UniformDistribution/Inline33.gif" style="height:66px; width:125px" />	(10)

and

			(11)
			(12)

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)
			(14)
			(15)

The first few are therefore given explicitly by

			(16)
			(17)
			(18)
			(19)

The central moments are given analytically by

			(20)
			(21)
			(22)

The first few are therefore given explicitly by

			(23)
			(24)
			(25)
			(26)

The mean, variance, skewness, and kurtosis excess are therefore

			(27)
			(28)
			(29)
			(30)

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REFERENCES:

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 531 and 533, 1987.