Difference between revisions of "Poisson distribution/RV"

Revision as of 20:49, 26 February 2018

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

There is no unique way to define a random variable, here is one way.

Let ( $[0,1]$ , $\mathcal{B}([0,1])$ , $\lambda$ ) 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, $\mathcal{P}(\mathbb{N}_0)$ ) as a sigma-algebra and X:[0,1]→N0 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.

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)

@@ Line 17: / Line 17: @@
 *** 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!} }}
-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>