Mean and Variance of the Discrete Uniform Distribution

Suppose $X$ is a discrete uniform random variable on the consecutive integers $\{a, a+1, a+2, \ldots, b\},$ so that each of the $n = b - a + 1$ values has probability $\dfrac{1}{n}.$

The mean of $X$ is the midpoint of the smallest and largest values:

$$\mu = E[X] = \dfrac{a+b}{2}$$

For example, if $X$ is the result of rolling a fair six-sided die, then $X$ is discrete uniform on $\{1, 2, 3, 4, 5, 6\},$ and

$$E[X] = \dfrac{1+6}{2} = \dfrac{7}{2} = 3.5.$$