Uniform Distribution

A uniform distribution assigns probabilities in proportion to a specified measure on a measurable set of finite positive measure. The continuous distribution on a finite interval, also known as a rectangular distribution, has constant probability density function on that interval and zero density outside it.

The probability density function and cumulative distribution function for a continuous uniform distribution on the interval , with , are

			(1)
			(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 distribution that is uniform, the characteristic function is

(5)

The value at is . If and , the characteristic function simplifies to

			(6)
			(7)

The moment-generating function is

			(8)
			(9)
			(10)

and

			(11)
			(12)

The moment-generating function is analytic at zero. Its expression for has a removable singularity there, so its derivatives at zero can be calculated by taking limits of the differentiated expression. 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)