Difference between revisions of "Mdm of a discrete distribution lemma"

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I may have found a useful transformation for calculating [[Mdm|Mdm's]] of distributions defined on {{M|\mathbb{Z} }} or a subset. I document my work so far below:
 
I may have found a useful transformation for calculating [[Mdm|Mdm's]] of distributions defined on {{M|\mathbb{Z} }} or a subset. I document my work so far below:
−
:* [[Notes:Mdm of a discrete distribution lemma]]
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:* [[Notes:Mdm of a discrete distribution lemma]]{{ProbMacros}}
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__TOC__
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==Statement==
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:: {{Notice|'''Notice: ''' - there are plans to generalise this lemma:- specifically to allow {{M|\lambda}} to take any real value (currently only non-negative allowed) and possibly also allow {{M|\alpha,\beta}} to be negative too}}
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Let {{M|\lambda\in\mathbb{R}_{\ge 0} }} and let {{M|\alpha,\beta\in\mathbb{N}_0}} such that {{M|\alpha\le\beta}}, let {{M|f:\{\alpha,\alpha+1,\ldots,\beta-1,\beta\}\subseteq\mathbb{N}_0\rightarrow\mathbb{R} }} be a [[function]], then we claim:
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* {{MM|\sum^\beta_{k\eq\alpha}\big\vert k-\lambda\big\vert f(k) \eq\sum^\gamma_{k\eq\alpha}(\lambda-k)f(k)  +\sum_{k\eq\gamma+1}^\beta (k-\lambda)f(k) }} where:
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** {{M|\gamma:\eq\text{Min}(\text{Floor}(\lambda),\beta)}}
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Note that {{M|\beta\eq\infty}} is valid for this expression (standard limits stuff, see ''[[sum to infinity]]'')
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==Applications to computing [[Mdm]]==
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Let {{M|X}} be a [[discrete random variable|''discrete'']] [[random variable]] defined on {{M|\{\alpha,\alpha+1,\ldots,\beta-1,\beta\}\subseteq\mathbb{N}_0}} (remember that {{M|\beta\eq\infty}} is valid and just turns the second sum into a limit), then:
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* define {{M|\lambda:\eq\E{X} }}
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* define {{M|f:k\mapsto \P{X\eq k} }}
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Recall the [[mdm]] of {{M|x}} is defined to be:
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* {{MM|\Mdm{X}:\eq \sum^\beta_{k\eq\alpha}\big\vert k-\E{X}\big\vert\ \P{X\eq k} }}
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It is easy to see that with the definitions substituted that:
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* {{MM|\sum^\beta_{k\eq\alpha}\big\vert k-\lambda\big\vert f(k)\eq\Mdm{X} }}
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==Proof==
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{{Requires proof|grade=A**|msg=Note follow
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* Initial notes [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 01:24, 22 January 2018 (UTC)
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*# A lot of work has been done in ''[[Notes:Mdm of a discrete distribution lemma]]'' and I've done each of the 4 cases individually ({{M|\alpha\eq\beta}}, {{M|\beta<\text{Floor}(\lambda)}}, {{M|\beta>\text{Floor}(\lambda)}} and {{M|\beta\eq\text{Floor}(\lambda)}} - but they need to be put together.
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*# There is a 5th case where {{M|\lambda<0}} is introduced
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*# I'd like to generalise this to {{M|\alpha,\beta\in\mathbb{Z} }} - generalising beyond {{M|\alpha,\beta}} being ''non-negative''}}
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==Notes==
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<references group="Note"/>
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==References==
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<references/>
 
{{Theorem Of|Statistics|Probability|Elementary Probability}}
 
{{Theorem Of|Statistics|Probability|Elementary Probability}}

Revision as of 01:24, 22 January 2018

I may have found a useful transformation for calculating Mdm's of distributions defined on [ilmath]\mathbb{Z} [/ilmath] or a subset. I document my work so far below:

  • Notes:Mdm of a discrete distribution lemma
    [ilmath]\newcommand{\P}[2][]{\mathbb{P}#1{\left[{#2}\right]} } \newcommand{\Pcond}[3][]{\mathbb{P}#1{\left[{#2}\!\ \middle\vert\!\ {#3}\right]} } \newcommand{\Plcond}[3][]{\Pcond[#1]{#2}{#3} } \newcommand{\Prcond}[3][]{\Pcond[#1]{#2}{#3} }[/ilmath]
    [ilmath]\newcommand{\E}[1]{ {\mathbb{E}{\left[{#1}\right]} } } [/ilmath][ilmath]\newcommand{\Mdm}[1]{\text{Mdm}{\left({#1}\right) } } [/ilmath][ilmath]\newcommand{\Var}[1]{\text{Var}{\left({#1}\right) } } [/ilmath][ilmath]\newcommand{\ncr}[2]{ \vphantom{C}^{#1}\!C_{#2} } [/ilmath]

Statement

Notice: - there are plans to generalise this lemma:- specifically to allow [ilmath]\lambda[/ilmath] to take any real value (currently only non-negative allowed) and possibly also allow [ilmath]\alpha,\beta[/ilmath] to be negative too

Let [ilmath]\lambda\in\mathbb{R}_{\ge 0} [/ilmath] and let [ilmath]\alpha,\beta\in\mathbb{N}_0[/ilmath] such that [ilmath]\alpha\le\beta[/ilmath], let [ilmath]f:\{\alpha,\alpha+1,\ldots,\beta-1,\beta\}\subseteq\mathbb{N}_0\rightarrow\mathbb{R} [/ilmath] be a function, then we claim:

  • [math]\sum^\beta_{k\eq\alpha}\big\vert k-\lambda\big\vert f(k) \eq\sum^\gamma_{k\eq\alpha}(\lambda-k)f(k) +\sum_{k\eq\gamma+1}^\beta (k-\lambda)f(k) [/math] where:
    • [ilmath]\gamma:\eq\text{Min}(\text{Floor}(\lambda),\beta)[/ilmath]

Note that [ilmath]\beta\eq\infty[/ilmath] is valid for this expression (standard limits stuff, see sum to infinity)

Applications to computing Mdm

Let [ilmath]X[/ilmath] be a discrete random variable defined on [ilmath]\{\alpha,\alpha+1,\ldots,\beta-1,\beta\}\subseteq\mathbb{N}_0[/ilmath] (remember that [ilmath]\beta\eq\infty[/ilmath] is valid and just turns the second sum into a limit), then:

  • define [ilmath]\lambda:\eq\E{X} [/ilmath]
  • define [ilmath]f:k\mapsto \P{X\eq k} [/ilmath]

Recall the mdm of [ilmath]x[/ilmath] is defined to be:

  • [math]\Mdm{X}:\eq \sum^\beta_{k\eq\alpha}\big\vert k-\E{X}\big\vert\ \P{X\eq k} [/math]

It is easy to see that with the definitions substituted that:

  • [math]\sum^\beta_{k\eq\alpha}\big\vert k-\lambda\big\vert f(k)\eq\Mdm{X} [/math]

Proof

Grade: A**
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Note follow
  • Initial notes Alec (talk) 01:24, 22 January 2018 (UTC)
    1. A lot of work has been done in Notes:Mdm of a discrete distribution lemma and I've done each of the 4 cases individually ([ilmath]\alpha\eq\beta[/ilmath], [ilmath]\beta<\text{Floor}(\lambda)[/ilmath], [ilmath]\beta>\text{Floor}(\lambda)[/ilmath] and [ilmath]\beta\eq\text{Floor}(\lambda)[/ilmath] - but they need to be put together.
    2. There is a 5th case where [ilmath]\lambda<0[/ilmath] is introduced
    3. I'd like to generalise this to [ilmath]\alpha,\beta\in\mathbb{Z} [/ilmath] - generalising beyond [ilmath]\alpha,\beta[/ilmath] being non-negative

Notes

References


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