Show the memoryless property is equivalent to other expressions.

Question

A random variable $X$ has memoryless property if $P( X \le s + t | X \gt s) = P(X \le t)$, $s, t \gt 0$.

Show that the property above is equivalent to

$P(X \gt s+ t | X \gt s) = P(X \gt t)$ and to $P(X \gt s +t) = P(X \gt s)P(X \gt t)$.

I really appreciate the help as I'm hopeless with proving these types of things. :)

Answer 1

The first is obvious: subtract 1 from both sides of your original equation.

For the second, note that:

$$\begin{align}P(X>s+t\mid X>s)P(X>s)&=P(X>t)P(X>s)\\ P(X>s+t, X>s)&=P(X>t)P(X>s)\\ P(X>s+t)&=P(X>t)P(X>s)\end{align}$$