View source for Deriving the exponential distribution from the time between event in a Poisson distribution

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{{ProbMacro}}
==Notes==
Let {{M|X\sim}}[[Poisson distribution|{{M|\text{Poi}(\lambda)}}]] for some {{M|\lambda\in\mathbb{R}_{>0} }}
* Here supposed that {{M|X}} models the number of events ''per unit time'' - although as with Poisson distribution - any continuum will do

Then:
* Suppose an event happens at {{M|t\eq t_0}}
** Let {{M|T}} be the [[random variable]] which is the time until the next event.
*** Let {{M|d\in\mathbb{R}_{>0} }} be given, so we can investigate {{M|\P{T>d} }}
*** We are interested in {{M|\mathbb{P}\big[\text{no events happening for time in }(t_0,t_0+d)\big]\eq\P{T>d} }}
**** Let {{M|X'\sim\text{Poi}(\lambda d)}} be used to model this interval
***** as if {{M|\lambda}} events are expected to occur per unit time, then {{M|\lambda d}} are expected to occur per unit {{M|d}} of time
***** It is easy to see that {{M|\mathbb{P}\big[\text{no events happening for time in }(t_0,t_0+d)\big]\eq\mathbb{P}\big[X'\eq 0\big]\eq e^{-\lambda d} }}
**** Thus {{M|\P{T>d}\eq e^{-\lambda d} }}
***** Or: {{M|\P{T\le d}\eq 1-\P{T>d}\eq 1-e^{-\lambda d} }}
** But this is what we'd see if {{M|T}} followed the [[exponential distribution]] with parameter {{M|\lambda d}}
*** {{M|\P{T\le d}\eq 1-e^{-\lambda d} }}
Thus we see the time between occurrences of events in a Poisson distribution is exponentially distributed, or [[memoryless property|memoryless]].

==Modifications==
Suppose instead {{M|t_0}} is the start time of the process rather than the last event time, how does this change things?

==Notes==
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==References==
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{{Theorem Of|Elementary Probability|Probability|Statistics}}