Recently one of my friends found a trivial solution:

$$Z_t = \frac{X}{P[X>t]}$$

Recently one of my friends found a trivial solution:

$$Z_t = \frac{X}{P[X>t]}$$

To see that it works:

$$P[Z_t > T] = \int_T^\infty \frac{\rho(x)}{P[X>t]} dx = \frac{1}{P[X>t]}\int_T^\infty \rho(x) dx = \frac{P[X>T]}{P[X>t]}$$