Uniform law

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Definition

Definition

Let $a, b \in \mathbb{R}$ with $a < b$. $X$ follows the uniform law on the interval $[a, b]$ if its density is :

$$f_X(x) = \frac{1}{b-a}\mathbb{1}_{[a, b]}(x).$$

We note $X \sim \mathcal{U}([a, b])$.

Results

Proposition : Uniform law's characteristic function

Let $X$ be a random variable. If $X \sim \mathcal U([a,b])$, the characteristic function of $X$ is

$$\phi_X(t) =\frac{e^{itb} - e^{ita}}{it(b-a)}.$$