# Discrete uniform distribution ¶

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A distribution, on an interval $[a,b]$, in which each value has the same probability, is a uniform distribution $U([a,b])$.

• The mass function is $\mathbb{P}(X=k) = \frac{1}{b-a+1}$
• $\mathbb{E}(X) = \ \frac{a+b}{2}$
• $\mathbb{V}(X) = \ \frac{(b-a)(b-a+2)}{12}$

Considering $[1,n]$, we would have

• The mass function is $\mathbb{P}(X=k) = \frac{1}{n}$
• $\mathbb{E}(X) = \ \frac{n+1}{2}$
• $\mathbb{V}(X) = \ \frac{n^2 - 1}{12}$