Chapter 3.2, Problem 4E

Let $F\left(x\right)$ be the cdf of the continuous-type random variable X. and assume that $F\left(x\right)=0$ for $x\le 0$ and $0<F\left(x\right)<1$ for $0<x$.

Prove that if $P\left(X>x+y|X>x\right)=P\left(X>y\right)$ for all $y>0$, then $F\left(x\right)=1-e^{-\lambda x},0<x$.

HINT: Show that $g\left(x\right)=1-F\left(x\right)$ satisfies the functional equation $g\left(x+y\right)=g\left(x\right)g\left(y\right)$, which implies that $g\left(x\right)=a^{cx}$.