One of my assignments was to show, that exponential distribution implies the memoryless property. I had no problem proving that, but as I was doing it, I was wondering, what do we get in reverse? So, what I mean is:
If we have $X>0$ as a random variable with continuous density function $f$, that satisfies:
$$P(X>t+s \mid X>s) = P(X>t)$$.
What can we say about $f$ and its form? Is there any kind of implication in that direction?
Best Answer
Note the LHS of the condition equals $$\frac{P(X>t+s)}{P(X>s)}, $$ since if $t$ and $s$ are positive then the event $(X>t+s,X>s)$ is the same as the event $(X>t+s)$. Writing $h(t):=P(X>t)$, what you have is the functional equation $$h(t+s)=h(s)h(t)\quad \text{for all $t,s>0$}$$ If we require $h$ to be continuous (actually it's enough for $h$ to be continuous at one point) then the only solution to the equation is the exponential function: $h(t):= \exp(\alpha t)$ for some $\alpha$. Imposing the restriction that $h(t)\to0$ as $t\to\infty$ means the coefficient $\alpha$ must be negative, and the solution is the exponential distribution.