Finding distribution function of random variable

Question

Given a random variable $\xi$, $\mathbb E_\xi = 1 $. It is known that $P(\xi > 2T | \xi > T) = P (\xi > T)$ $\forall T> 0$. How to find $ F_\xi$?

Answer 1

Let $\overline{F}_\xi(T) := 1 - F_\xi(T)$. $$\overline{F}_\xi(T) = P(\xi > T) = P(\xi > 2T \mid \xi > T) = \frac{P(\xi > 2T)}{P(\xi > T)} = \frac{\overline{F}_\xi(2T)}{\overline{F}_\xi(T)}$$ Thus, $$(\overline{F}_\xi(T))^2 = \overline{F}_\xi(2T), \forall T > 0.$$ If $\overline{F}_\xi$ is continuous, then this necessarily implies $\overline{F}_\xi(T) = e^{-\lambda T}$ and thus $F_\xi(T) = 1 - e^{-\lambda T}$ for some real $\lambda$. Using $\mathbb{E} \xi = 1$ implies $\lambda = 1$.