You have two lightbulbs for a particular lamp. Let $X=$ the lifetime of the first light bulb and $Y =$ the lifetime of the secomd bulb. We turn on the bulbs successively. Suppose that $X$ and $Y$ are independent and $X$~$\operatorname {Exp}(\lambda)$, $Y$~$\min(\operatorname {Exp}(\mu),d)$, where $d$ is a positive constant.
I need to find $\operatorname F_{X|Z}(x|z)$ and $\operatorname f_{X|Z}(x|z)$, where $Z=X+Y$.
I started with $\operatorname P(X+Y \leq t)$ =
a) $t<d$: =$\int_{0}^{t}\int_{0}^{t-y}\lambda\mu e^{-\lambda x-\mu y}dxdy$
b) $t\geq d$: =$\operatorname P(Y \leq d) \operatorname P(X\leq t-d) + \int_{0}^{d}\int_{t-d}^{t-y}\lambda\mu e^{-\lambda x-\mu y}dxdy$ = $\int_{0}^{t-d}\lambda e^{-\lambda x}dx + \int_{0}^{d}\int_{t-d}^{t-y}\lambda\mu e^{-\lambda x-\mu y}dxdy$
Now, I don't know how to proceed. Can you help me?