Expectation of Discrete Uniform Distribution

Theorem
Let $X$ be a discrete random variable with the discrete uniform distribution with parameter $n$.

Then the expectation of $X$ is given by:
 * $\expect X = \dfrac {n + 1} 2$

Proof
From the definition of expectation:
 * $\ds \expect X = \sum_{x \mathop \in \Omega_X} x \map \Pr {X = x}$

Thus: