Actuarial Exam Question

#1
Hello All,
I am studying for the actuarial exam, and I have come across a question which I do not understand the answer to:

A device contains two circuits. The second circuit is a backup for the first, so the second is used only when the first has failed. The device fails when and only when the second circuit fails. Let X and Y be the times at which the first and second circuits fail, respectively. X and Y have the joint probability density function:

f(x,y)=6Exp[-x]Exp[-2y], for 0<x<y<Infinity

What is the expected time at which the device fails?

The solution calculates the marginal distribution of Y, and then the expected value of Y from the marginal distribution, and that is the answer. Why do we not take into account the expected value of X? What am I missing? Thanks!
 

Mean Joe

TS Contributor
#2
When you find the marginal of Y (by integrating the double integral with respect to X), that is when you are taking account of X. Hope this answers your question.