Exponential distribution

Definition

Let $\lambda\in\mathbb{R}_{\ge 0}$ be given, and let $X\sim\text{Exp}(\lambda)$ be an exponentially distributed random variable. Then:

• the probability density function, $f:\mathbb{R}_{\ge 0}\rightarrow\mathbb{R}_{\ge 0}$ is given as follows:
  • $f:x\mapsto \lambda e^{-\lambda x}$, from this we can obtain:
• the cumulative distribution function, $F:\mathbb{R}_{\ge 0}\rightarrow[0,1]\subseteq\mathbb{R}$, which is:
  • $F:x\mapsto 1-e^{-\lambda x}$
    • The proof of this is claim 1 below

The exponential distribution has the memoryless property