DES - Definition of 'Extended Memory-Less Property'

Extended Memory-Less Property:

Let $X$, $Y$ be independent variables such that:

• $X\sim Exp\left(\frac{1}{\lambda }\right)$
• $f_Y(y)=0\text{for}y\le 0$
  • where: $f_Y$ denotes the pdf of $Y$.
• Moreover, let $s\ge 0$.

Then the Extended Memory-Less Property enunciate that:

$$P(X>Y+s\mid X>Y)=P(X>s)$$

And using the probability of the complement of an event:

$$P(X\le Y+s\mid X>Y)=P(X\le s)$$

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Demonstration

DES - Demonstration of the Extended Memory-Less Property

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