How to calculate the probability of $\Pr(U>t\mid U<Z)$ for two independent exponential random variables $Z$ and $U$

Question

Suppose we have two independent random variables $Z$ and $U$ with exponential probability distributions with two rates $\lambda_z$ and $\lambda_u$, respectively. Can we write as follows:

$\Pr(U>t\mid U<Z)=\frac{\Pr(U>t \And Z>U)}{\Pr(U<Z)}=\frac{\Pr(t<U<Z)}{\Pr(U<Z)}$

If yes, how to calculate $\Pr(t<U<Z)$?

Answer 1

Yes, you can write it that way.

$P(t<U<Z)=\int_t^{\infty} \int_u^{\infty} \lambda_u e^{-\lambda_u v}\lambda_z e^{-\lambda_z v}dvdu$ and $P(U<Z)=\int_0^{\infty} \int_u^{\infty} \lambda_u e^{-\lambda_u v}\lambda_z e^{-\lambda_z v}dvdu$.