PROBABILITIES Course

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

  • $a=0$
  • $b=1$

For more information, here is the wiki.

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

@ X = F_Y(Y) \sim U([0,1]) @

Using the Probability integral transform (PIP).