DES - Demonstration of the Extended Memory-Less Property

Arguments:

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

Also for the hypothesis we have that $Y$ is always positive and so is $X$ (from the definition of the pdf of the exponential distribution). So we are interested only on the first quadrant, where it applies that $x>y+s$

We can integrate $x$ from $y+s$ instead of $0$ because $x=0$ for $x\le y+s$ as we can see from the graph before.

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