I got it finally. But here is a brief note about it
E[t_2-t_1|t_2>t_1] = E[(t_2-t_1)*I_{\{t_2>t_1\}}] = \int_0^{\infty} \int_{t_1}^{\infty}(t_2-t_1)\lambda_1 \lambda_2 e^{-\lambda_1 t_1} e^{-\lambda_2 t_2} dt_2 dt_1
This nicely gives the answer as...
I have been stuck at this calculation. There are two exponential distributions X and Y with mean 6 and 3 respectively. We need to find
E[y-x|y>x]
I keep getting it negative, which is clearly wrong. Anybody wants to try it?