Difference between revisions of "Poisson distribution/RV"

Latest revision as of 20:59, 26 February 2018

Definition

As a formal random variable

Caveat:$\lambda$ 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.

• Let $\big($$[0,1]$$,\ $$\mathcal{B}([0,1])$$,\ $$\lambda$$\big)$ be a probability space - which itself could be viewed as a rectangular distribution's random variable
  • Let $\lambda\in\mathbb{R}_{>0}$ be given, and let $X\sim\text{Poi}(\lambda)$
    • Specifically consider $\big(\mathbb{N}_0,\ $$\mathcal{P}(\mathbb{N}_0)$$\big)$ as a sigma-algebra and $X:[0,1]\rightarrow\mathbb{N}_0$ by:
      • $X:x\mapsto\left\{\begin{array}{lr}0&\text{if }x\in[0,p_1)\\1 & \text{if }x\in[p_1,p_2)\\ \vdots & \vdots \\ k & \text{if }x\in[p_k,p_{k+1})\\ \vdots & \vdots \end{array}\right.$ for $$p_1:\eq e^{-\lambda} \frac{\lambda^1}{1!}$$ and $p_k:\eq p_{k-1}+e^{-\lambda}\frac{\lambda^k}{k!}$

Giving the setup shown on the left.

TODO: TODO:

• Surely it should be $[0,1)$ and $\mathcal{B}\big([0,1)\big)$ for this to work? Alec (talk) 20:49, 26 February 2018 (UTC)