Exponentially distributed random variable inquality (using memoryless property)

Question

This is my first post :) So given a exponentially distributed random variable $X$, how do I show that for $t,s$ positive integers and $u\in [0,s]$: $$ P(X>t+s-u)=P(X>t)P(X>s-u) $$ I know that I need to use the memoryless property, but I am stuck at how to do that? I am a bit confused how to get to the product.

Answer 1

HINT

Note that since $0 \le u \le s$ we have $s-u > 0$ so letting $v = s-u$ you must show that $$ \mathbb{P}[X > t+v] = \mathbb{P}[X>t] \cdot \mathbb{P}[X > v] $$

Consider conditioning on $X>t$. We have $$ \mathbb{P}[X > t+v] = \mathbb{P}[X > t+v, X > t] = \mathbb{P}[X > t] \cdot \mathbb{P}[X > t+v | X > t] $$ Can you now use the memoryless property to simplify?