Why does the expected value $E(X)$ of a Discrete Uniform Variable, X, $= \frac{N+1}{2}$ and the variance formula $Var(X) = \frac{N^2-1}{12}$?

Question

My textbook gives the above as formulas, but does not give an derivation or proof for those formulas. Can someone provide me with a explanation or a resource that explains where these formulas come from so I can better understand these formulas? Thank You In Advance.

Answer 1

Suppose $\Omega = \{1, 2, \cdots N\}$, and $X$ is uniform on it, i.e, $\mathbb{P}(X = x) = 1/N$ for every $x \in \Omega$.

Then $\mathbb{E}[X] = \Sigma_{n \leq N}n\cdot \mathbb{P}(X = n) = \Sigma_{n \leq N} \frac{n}{N} = \frac{\Sigma_{n \leq N}}{N} = \frac{N(N+1)/2}{N} = \frac{N+1}{2}$.

Can you figure out how to do $Var[X]$? It's basically the same idea but a with bit more tedious calculation.