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{{ProbMacros}}{{Stub page|grade=A**|msg={{Warning|The notation and terminology used here was developed by me for a project, it has not been researched '''yet'''.}}}}

Before we start I must point out something which may without deeper thought appear to be a Poisson mixture, recall that:
* [[Addition of Poisson distributions|The addition of two Poisson distributions is itself a Poisson distribution]], and
* {{XXX|Link here}} {{C|If we have a Poisson distribution and each of its events being noticed i.i.d and BORV(p) then the observed process is also a Poisson distribution}}
** That is, let {{M|X\sim\text{Poi}(\lambda)}} and let {{M|(X_i)_{i\in\mathbb{N} } }} be the events detected by this "process". Let {{M|(D_i)_{i\in\mathbb{N} } }} be the detection of each event, {{iid}} and {{M|\forall i\in\mathbb{N}\big[D_i\sim\text{Borv}(p)\big]}} then {{M|Y\sim\text{Poi}(p\lambda)}} is the distribution of the {{M|(D_i)_{i\in\mathbb{N} } }} events.
*** This is not phrased very well.
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==Definition==
Let {{M|(\lambda_i)_{i\eq 1}^n\subseteq\mathbb{R}_{\ge 0} }} be given, and let {{M|\big(X_i\sim\text{Poi}(\lambda_i)\big)_{i\eq 1}^n}} be a sequence of {{M|\mathbb{N} }}-valued [[Poisson distribution|Poisson distributed random variables]], then

Let {{M|\mathcal{C} }} be an {{M|N}}-valued [[random variable]] where {{M|N:\eq\{1,2,3,\ldots,n\}\subseteq\mathbb{N} }}¸ next
* Define {{M|X}} to be an {{M|\mathbb{N} }}-valued random variable as follows:
** for any {{M|k\in\mathbb{N} }}, define: {{MM|\P{X\eq k}:\eq \sum^n_{i\eq 1}\Big(\P{\mathcal{C}\eq i}\cdot\Pcond{X_i \eq k}{\mathcal{C}\eq i}\Big) }} - which is just a standard composition {{XXX|Is this [[law of total probability]]?}}
***  Notice: {{MM|\P{X\eq k}\eq\sum^n_{i\eq 1}\Big(\P{\mathcal{C}\eq i}\cdot\P{X_i\eq m}\Big)}} {{XXX|Justification?}}
***: {{MM|\eq \sum^n_{i\eq 1}\left(\P{\mathcal{C}\eq i}\cdot\frac{e^{-\lambda_i}\cdot \lambda_i^k}{k!} \right)}}
***: {{MM|\eq \frac{1}{k!}\sum^n_{i\eq 1}\Big(e^{-\lambda_i}\cdot\P{\mathcal{C}\eq i}\cdot \lambda_i^k\Big) }}

The idea is that we first let {{M|(x_i)_{i\eq 1}^n}} be a sample from the {{M|(X_i)_{i\eq 1}^n}} random variables. Let {{M|c}} be a sample of {{M|\mathcal{C} }}, then {{M|x}}, our sample of {{M|X}} is:
* {{M|x:\eq x_c}}
===Special case: {{M|n\eq 2}}===
Let {{M|\P{\mathcal{C}\eq 1}:\eq p}}, thus {{M|\P{\mathcal{C}\eq 2}\eq 1-p}}, then:
* {{MM|\P{X\eq k}\eq\frac{1}{k!}\Big(p e^{-\lambda_1} \lambda_1^k+(1-p)e^{-\lambda_2}\lambda_2^k\Big)}}
Finding the parameters would be an [[optimisation problem]] (minimise error with {{M|3}} parameters here, {{M|\lambda_1,\ \lambda_2\in\mathbb{R}_{\ge 0} }} and {{M|p\in[0,1]\subseteq\mathbb{R} }}
==Notes==
<references group="Note"/>
{{Definition|Probability|Statistics}}