Difference between revisions of "Exponential distribution/Definition"

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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 on the exponential distribution page

The exponential distribution has the memoryless property^{[Note 1]}

Notes

1. Jump up ↑ Furthermore, the memoryless property characterises the exponential distribution, that is a distribution has the memoryless property if and only if it is a member of the exponential distribution family, i.e. an exponential distribution for some $\lambda\in\mathbb{R}_{>0}$

References