Exponential distribution/Definition

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use Rice's [ilmath]\mathbb{P} [/ilmath] book. Page 48

Definition

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

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

The exponential distribution has the memoryless property[Note 1]

Notes

  1. 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 [ilmath]\lambda\in\mathbb{R}_{>0} [/ilmath]

References