Discrete Uniform Distribution

The discrete uniform distribution assigns equal probability to each element of a finite nonempty set. It is also known as the "equally likely outcomes" distribution. Let have elements, where is a positive integer, and let be the event consisting of its th element. Then

so using gives

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

for , ..., .

The discrete uniform distribution is implemented in the Wolfram Language as DiscreteUniformDistribution[1, N] for .

Its moment-generating function is

At , use the limiting value . The moments about 0 are

so

and the moments about the mean are

The mean, variance, skewness, and kurtosis excess are

For , the variance is zero and the skewness and kurtosis excess are undefined. The displayed formulas for these two standardized quantities require .

The mean deviation for the discrete uniform distribution on 1, 2, ..., is given by

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

Similarly, for even,

The complete solution is therefore

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