Discrete Uniform Distribution

A random variable $X$ has a discrete uniform distribution iff each possible value of $X$ is equally likely

f (x) = \frac{1}{k} for x = x_{1}, x_{2}, \dots, x_{k}

where no 2 values are the same

\begin{aligned} E (X) & = \sum_{x} x f (x) \\ = \sum_{x = 1}^{n} x \frac{1}{n} \\ = \frac{1}{n} \frac{n (n + 1)}{2} \\ μ & = \frac{n + 1}{2} \end{aligned}

\begin{aligned} V a r (X) & = E (X^{2}) - [E (X)]^{2} \\ = \frac{1}{n} \sum_{x = 1}^{n} x^{2} - {(\frac{n + 1}{2})}^{2} \\ = \frac{1}{n} \frac{n (n + 1) (2 n + 1)}{6} - {(\frac{n + 1}{2})}^{2} \\ σ^{2} & = \frac{n^{2} - 1}{12} \end{aligned}