Difference between revisions of "Exponential distribution/Definition"
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Revision as of 16:11, 3 November 2017
Grade: B
This page requires references, it is on a to-do list for being expanded with them.
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use Rice's [ilmath]\mathbb{P} [/ilmath] book. Page 48
Contents
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:
- [ilmath]F:x\mapsto 1-e^{\lambda x} [/ilmath]
- The proof of this is claim 1 on the exponential distribution page
- [ilmath]F:x\mapsto 1-e^{\lambda x} [/ilmath]
The exponential distribution has the memoryless property[Note 1]
Notes
- ↑ 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
