Difference between revisions of "Exponential distribution/Definition"
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(Created page with "<noinclude> {{Requires references|use Rice's {{M|\mathbb{P} }} book. Page 48|grade=B}} __TOC__ ==Definition== </noinclude>Let {{M|\lambda\in\mathbb{R}_{\ge 0} }} be given, and...") |
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{{Requires references|use Rice's {{M|\mathbb{P} }} book. Page 48|grade=B}} | {{Requires references|use Rice's {{M|\mathbb{P} }} book. Page 48|grade=B}} | ||
__TOC__ | __TOC__ | ||
| + | ==Using this page== | ||
| + | Set {{c|home}} to something when using this page to change the "the proof of this is ''claim 1'' message" | ||
==Definition== | ==Definition== | ||
</noinclude>Let {{M|\lambda\in\mathbb{R}_{\ge 0} }} be given, and let {{M|X\sim\text{Exp}(\lambda)}} be an ''exponentially distributed'' [[random variable]]. Then: | </noinclude>Let {{M|\lambda\in\mathbb{R}_{\ge 0} }} be given, and let {{M|X\sim\text{Exp}(\lambda)}} be an ''exponentially distributed'' [[random variable]]. Then: | ||
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** {{M|f:x\mapsto \lambda e^{-\lambda x} }}, from this we can obtain: | ** {{M|f:x\mapsto \lambda e^{-\lambda x} }}, from this we can obtain: | ||
* the [[cumulative distribution function]], {{M|F:\mathbb{R}_{\ge 0}\rightarrow[0,1]\subseteq\mathbb{R} }}, which is: | * the [[cumulative distribution function]], {{M|F:\mathbb{R}_{\ge 0}\rightarrow[0,1]\subseteq\mathbb{R} }}, which is: | ||
| − | ** {{M|F:x\mapsto 1-e^{\lambda x} }} | + | ** {{M|F:x\mapsto 1-e^{-\lambda x} }} |
*** The proof of this is '''claim 1''' {{#if:{{{home|}}}|below|on the [[exponential distribution]] page}} | *** The proof of this is '''claim 1''' {{#if:{{{home|}}}|below|on the [[exponential distribution]] page}} | ||
Latest revision as of 01:27, 16 March 2018
Grade: B
This page requires references, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
The message provided is:
The message provided is:
use Rice's [ilmath]\mathbb{P} [/ilmath] book. Page 48
Contents
Using this page
Set home to something when using this page to change the "the proof of this is claim 1 message"
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
