# Continuous 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 density function is $f_X(x) = \frac{1}{b-a}$
• $\mathbb{E}(X) = \ \frac{a+b}{2}$
• $\mathbb{V}(X) = \ \frac{(b-a)^2}{12}$

## Standard uniform distribution ¶

We are calling standard uniform distribution/loi uniforme standard a uniform distribution where

• $a=0$
• $b=1$

Let $F_Y(y)$ be the cumulative distribution function of the continuous random variable Y. Then $X = F_Y(y)$ can follow a standard uniform distribution