iscrete Uniform Distribution

The discrete uniform distribution is also known as the "equally likely outcomes" distribution. Letting a set  have  elements, each of them having the same probability, then

			(1)
			(2)
			(3)
			(4)

so using  gives

(5)

Restricting the set  to the set of positive integers 1, 2, ..., , the probability distribution function and cumulative distributions function for this discrete uniform distribution are therefore

			(6)
			(7)

for , ..., .

The discrete uniform distribution is implemented in the Wolfram Language as DiscreteUniformDistribution[n].

Its moment-generating function is

			(8)
			(9)
			(10)
			(11)

The moments about 0 are

(12)

so

			(13)
			(14)
			(15)
			(16)

and the moments about the mean are

			(17)
			(18)
			(19)

The mean, variance, skewness, and kurtosis excess are

			(20)
			(21)
			(22)
			(23)

The mean deviation for a uniform distribution on  elements is given by

(24)

To do the sum, consider separately the cases of  odd and  even. For  odd,

			(25)
			(26)
			(27)
			(28)

Similarly, for  even,

			(29)
			(30)
			(31)
			(32)

The complete solution is therefore

(33)

For , 2, ..., the first few values are 0, 1/2, 2/3, 1, 6/5, 3/2, 12/7, ... (OEIS A086111 and A086112).

REFERENCES:

Sloane, N. J. A. Sequences A086111 and A086112 in "The On-Line Encyclopedia of Integer Sequences."