I am reviewing for a probability and statistics class. I am stuck on a problem despite repeated attempts. (THIS IS NOT HOMEWORK!)
The questions is: Consider an electronic system with two components. Suppose the system is such that one component is on the reserve and it is activated only if the other component fails. The system fails if and only if both the components fail. Let $X$ and $Y$ denote the life times of these components. Suppose $(X,Y)$ has the joint probability density function-
$$\begin{matrix} f(x,y) = \lambda^{2}e^{-\lambda(x+y)} & & \text{for } x \geqslant 0 \text{ and } y \geqslant 0. \end{matrix}$$
I tried to solve the problem by solving:
$$ f(x,y) = \int_0^\infty \int_{500-y}^\infty \lambda^{2}e^{-\lambda(x+y)}dxdy. $$
But this reduces to:
$$ f(x,y) = \int_0^\infty \lambda^{2}e^{-\lambda y}e^{-\lambda (500-y)}dy, $$
which becomes... $$ f(x,y) = \int_0^\infty \lambda^{2}e^{-\lambda y}e^{\lambda y}e^{-\lambda 500}dy\,=\infty. $$ Can someone point out what I'm overlooking? I considered changing variables to $Z = X+Y$ but this involves the Jacobian and I don't think the problem was intended to be quite that involved. (https://en.wikipedia.org/wiki/Probability_density_function#Multiple_variables)