Memoryless property

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This needs to be on more formal groundings. Also is there even a point? For the continuous case memoryless if and only if it's the exponential distribution - what about the discrete!

[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]

Definition

Let [ilmath]T[/ilmath] be a random variable representing time until something happens. Suppose that this thing has already survived until [ilmath]T\eq t_0[/ilmath].

• We say [ilmath]T[/ilmath] is memoryless or has the memoryless property if:
  • [ilmath]\forall d\in\mathbb{R}_{>0}\Big[\Pcond{T>t_0+d}{T>t_0}\eq \P{T>d}\Big][/ilmath]
    • That is to say the chance of it surviving a further [ilmath]d[/ilmath] time is the same as if it were a new item.

This makes the memoryless property:

• [ilmath]\forall t_0\in\mathbb{R}_{\ge 0}\forall d\in\mathbb{R}_{>0}\Big[\Pcond{T>t_0+d}{T>t_0}\eq \P{T>d}\Big][/ilmath]

Generalisation to a discrete random variable

Warning:Alec is investigating this, but what is proved is fine

I suspect that if [ilmath]X[/ilmath] is a discrete random variable defined on say [ilmath]\mathbb{N}_0[/ilmath] then we can phrase the memoryless property as:

• [ilmath]\forall m\in\mathbb{N}_{\ge 0}\forall n\in\mathbb{N}_{>0}\Big[\Pcond{X>m+n}{T>m}\eq \P{T>n}\Big][/ilmath]