Characterization of Memorylessness and the Residual Life Paradox

Part (a): Let $X$ be a continuous, positive random variable satisfying $P(X > s + t \mid X > s) = P(X > t)$ for all $s, t \geq 0$. Prove that $X$ must be exponentially distributed.

Part (b): A lightbulb's lifetime $L$ has CDF $F(t) = 1 - \frac{1}{2}e^{-t} - \frac{1}{2}e^{-3t}$ for $t \geq 0$ (a mixture of $\operatorname{Exp}(1)$ and $\operatorname{Exp}(3)$). You arrive at a uniformly random time and observe the bulb currently in use. Let $R$ be the residual lifetime of that bulb. Show that $E[R] > E[L]$ and compute both values. Explain why memorylessness breaks down and causes this paradox.