Difference between revisions of "Poisson distribution/RV"
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| + | : {{Caveat|{{M|\lambda}} here is used to denote 2 things}} - the parameter to the Poisson distribution, and the restriction of the 1 dimensional [[Lebesgue measure]] to some region of interest. | ||
There is no unique way to define a [[random variable]], here is one way. | There is no unique way to define a [[random variable]], here is one way. | ||
| − | |||
* Let {{M|\big(}}[[closed interval|{{m|[0,1]}}]]{{M|,\ }}[[Borel sigma-algebra of the real line|{{M|\mathcal{B}([0,1])}}]]{{M|,\ }}[[Lebesgue measure|{{M|\lambda}}]]{{M|\big)}} be a [[probability space]] - which itself could be viewed as a [[rectangular distribution|rectangular]] distribution's [[random variable]] | * Let {{M|\big(}}[[closed interval|{{m|[0,1]}}]]{{M|,\ }}[[Borel sigma-algebra of the real line|{{M|\mathcal{B}([0,1])}}]]{{M|,\ }}[[Lebesgue measure|{{M|\lambda}}]]{{M|\big)}} be a [[probability space]] - which itself could be viewed as a [[rectangular distribution|rectangular]] distribution's [[random variable]] | ||
** Let {{M|\lambda\in\mathbb{R}_{>0} }} be given, and let {{M|X\sim\text{Poi}(\lambda)}} | ** Let {{M|\lambda\in\mathbb{R}_{>0} }} be given, and let {{M|X\sim\text{Poi}(\lambda)}} | ||
*** Specifically consider {{M|\big(\mathbb{N}_0,\ }}[[power set|{{M|\mathcal{P}(\mathbb{N}_0)}}]]{{M|\big)}} as a [[sigma-algebra]] and {{M|X:[0,1]\rightarrow\mathbb{N}_0}} by: | *** Specifically consider {{M|\big(\mathbb{N}_0,\ }}[[power set|{{M|\mathcal{P}(\mathbb{N}_0)}}]]{{M|\big)}} as a [[sigma-algebra]] and {{M|X:[0,1]\rightarrow\mathbb{N}_0}} by: | ||
**** {{M|X:x\mapsto\left\{\begin{array}{lr}0&\text{if }x\in[0,p_1)\\1 & \text{if }x\in[p_1,p_2)\\ \vdots & \vdots \\ k & \text{if }x\in[p_k,p_{k+1})\\ \vdots & \vdots \end{array}\right.}} for {{MM|p_1:\eq e^{-\lambda} \frac{\lambda^1}{1!} }} and {{M|p_k:\eq p_{k-1}+e^{-\lambda}\frac{\lambda^k}{k!} }} | **** {{M|X:x\mapsto\left\{\begin{array}{lr}0&\text{if }x\in[0,p_1)\\1 & \text{if }x\in[p_1,p_2)\\ \vdots & \vdots \\ k & \text{if }x\in[p_k,p_{k+1})\\ \vdots & \vdots \end{array}\right.}} for {{MM|p_1:\eq e^{-\lambda} \frac{\lambda^1}{1!} }} and {{M|p_k:\eq p_{k-1}+e^{-\lambda}\frac{\lambda^k}{k!} }} | ||
| − | Giving the setup shown on the left. | + | Giving the setup shown on the left.<noinclude> |
| + | =={{XXX|TODO:}}== | ||
| + | * Surely it should be {{M|[0,1)}} and {{M|\mathcal{B}\big([0,1)\big)}} for this to work? [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 20:49, 26 February 2018 (UTC) | ||
| + | </noinclude> | ||
Latest revision as of 20:59, 26 February 2018
Definition
As a formal random variable
- Caveat:[ilmath]\lambda[/ilmath] here is used to denote 2 things - the parameter to the Poisson distribution, and the restriction of the 1 dimensional Lebesgue measure to some region of interest.
There is no unique way to define a random variable, here is one way.
- Let [ilmath]\big([/ilmath][ilmath][0,1][/ilmath][ilmath],\ [/ilmath][ilmath]\mathcal{B}([0,1])[/ilmath][ilmath],\ [/ilmath][ilmath]\lambda[/ilmath][ilmath]\big)[/ilmath] be a probability space - which itself could be viewed as a rectangular distribution's random variable
- Let [ilmath]\lambda\in\mathbb{R}_{>0} [/ilmath] be given, and let [ilmath]X\sim\text{Poi}(\lambda)[/ilmath]
- Specifically consider [ilmath]\big(\mathbb{N}_0,\ [/ilmath][ilmath]\mathcal{P}(\mathbb{N}_0)[/ilmath][ilmath]\big)[/ilmath] as a sigma-algebra and [ilmath]X:[0,1]\rightarrow\mathbb{N}_0[/ilmath] by:
- [ilmath]X:x\mapsto\left\{\begin{array}{lr}0&\text{if }x\in[0,p_1)\\1 & \text{if }x\in[p_1,p_2)\\ \vdots & \vdots \\ k & \text{if }x\in[p_k,p_{k+1})\\ \vdots & \vdots \end{array}\right.[/ilmath] for [math]p_1:\eq e^{-\lambda} \frac{\lambda^1}{1!} [/math] and [ilmath]p_k:\eq p_{k-1}+e^{-\lambda}\frac{\lambda^k}{k!} [/ilmath]
- Specifically consider [ilmath]\big(\mathbb{N}_0,\ [/ilmath][ilmath]\mathcal{P}(\mathbb{N}_0)[/ilmath][ilmath]\big)[/ilmath] as a sigma-algebra and [ilmath]X:[0,1]\rightarrow\mathbb{N}_0[/ilmath] by:
- Let [ilmath]\lambda\in\mathbb{R}_{>0} [/ilmath] be given, and let [ilmath]X\sim\text{Poi}(\lambda)[/ilmath]
Giving the setup shown on the left.
