I'm new to math stack exchange, and I have a question about an exercise that I'm doing. I have the following p.d.f. $$ f(x; \theta, \lambda) = \lambda \theta^\lambda x^{-(\lambda + 1)} $$ Where $x>0$, $\lambda>0$, $\theta > 0$. Let $\psi > \theta$, how do I determine the p.d.f. of $X$ conditioned to the event $X>\psi$?
How I have proceeded:
We know that $$ P(X \leq t | X>\psi)= \frac{P(X \leq t \; \cap \; X > \psi)}{P(X > \psi)} $$ Integrating the p.d.f. with the appropriate bounds, yields the following: $$ P(X > \psi) = 1 - P(X \leq \psi) = \theta^\lambda\psi^{-\lambda} $$ Lastly, $$ P(X \leq t \; \cap \; X > \psi) = \theta^\lambda (\psi^{-\lambda} - t^{-\lambda}) $$ Therefore, the c.d.f. will be equal to: $$ P(X \leq t | X>\psi)= \frac{\psi^{-\lambda} - t^{-\lambda}}{\psi^{-\lambda}} $$ Differentiating w.r.t. $t$ ultimately yields the p.d.f.:
$$ f(t) = \frac{\psi^{-\lambda} + \lambda t^{-\lambda - 1}}{\psi^{-\lambda}} $$
Is my solution correct?