Poisson distribution/RV

Definition

As a formal random variable

$\xymatrix{ & [0,1] \ar[r]^X & \mathbb{N}_0 \\ & \mathcal{B}([0,1]) \ar[dl]_-{\lambda} & \mathcal{P}(\mathbb{N}_0) \ar[l]_-{X^{-1} } \ar@{-->}@/^1em/[dll]^-{\mathbb{P}:\eq \lambda\circ X^{-1} } \\ \mathbb{R} & & }$
Situation for our RV

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.