Let $W$ a random variable such that:
$$F_W(w)=1-e^{-\lambda(w-t)}-\lambda(w-t)e^{w-t} \text{ for } t\le w$$
Compute $E[W]$
I think that it's correct to use the right tail formula for expectation:
$$E[W]=\int_t^\infty(1-F_W(w))dw$$ computing that integral I obtain that $E[W]=\frac{2}{\lambda}$, but the answer given is $E[W]=t+\frac{2}{\lambda}$
It's incorrect to use the right tail formula in this case?