Exponential law

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Definition

Definition

Let $\lambda > 0$. $X$ follows the exponential law with parameter $\lambda$ if its density is :

$$f_X(x) = \lambda e^{-\lambda x}\mathbb{1}_{\mathbb{R}^+}(x).$$

We note $X \sim \mathcal{E}(\lambda)$

Results

Proposition : Exponential law's characteristic function

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

$$\phi_X(t) = \frac{\lambda}{\lambda - it}$$