I wnated to know if I have solved this problem right or if I have done some errors.
Can someone help me?
Let $X$ be an exponentially distributed random variable, i.e. for the parameter $λ > 0$ the density is given by $f(x) =λe^{−λx}\mathbb{1}_{[0,∞)}(x)$. For $t > 0$ determine the conditional distribution $P[X > s + t | X > t]$, $s ≥ 0$.
So I calculated as follows:
$P[X > s + t | X > t]=\frac{P[X > s + t, X > t]}{P[X >t]}=\frac{P[X > s + t]}{P[X >t]}\\ =\frac{1-P[X \leq s + t]}{1-P[X \leq t]}=\frac{1-(1-e^{−λ(s+t)})}{1-(1-e^{−λ(t)})}\\ =\frac{e^{−λ(s+t)}}{e^{−λt}}=e^{-λs}$