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. | ||
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* 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\{0if x∈[0,p1)1if x∈[p1,p2)⋮⋮kif x∈[pk,pk+1)⋮⋮\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\{0if x∈[0,p1)1if x∈[p1,p2)⋮⋮kif x∈[pk,pk+1)⋮⋮\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:λ 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 ([0,1], B([0,1]), λ) be a probability space - which itself could be viewed as a rectangular distribution's random variable
- Let λ∈R>0 be given, and let X∼Poi(λ)
- Specifically consider (N0, P(N0)) as a sigma-algebra and X:[0,1]→N0 by:
- X:x↦{0if x∈[0,p1)1if x∈[p1,p2)⋮⋮kif x∈[pk,pk+1)⋮⋮ for p1:=e−λλ11! and pk:=pk−1+e−λλkk!
- Specifically consider (N0, P(N0)) as a sigma-algebra and X:[0,1]→N0 by:
- Let λ∈R>0 be given, and let X∼Poi(λ)
Giving the setup shown on the left.
