Since a uniform distribution's probability function is defined by a mathematical function (as opposed to a list of values), we can find its mean and variance by summing series.
Uniform distribution's mean and variance
If \(X \sim \UniformDistn(a, b) \), its mean and variance are
\[ \begin {align} E[X] & = \frac {a+b} 2 \\ \Var(X) & = \frac { (b - a + 1)^2 - 1} {12} \end {align} \](Proved in full version)