Cauchy's law

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Definition

Definition

Let $c > 0$. $X$ follows the Cauchy's law with the parameter $c$ if its density is :

$$f_X(x) = \frac{1}{\pi}\frac{c}{c^2+x^2}.$$

We note $X \sim \mathcal{C}(c)$.

Results

Proposition : Cauchy law's characteristic function

Let $X$ be a random variable. If $X \sim \mathcal C(c)$, the characteristic function of $X$ is

$$\phi_X(t) =e^{-c|t|}.$$