*Question:*

A device containing two key components fails when, and only when, both components fail. The lifetimes, T1 and T2, of these components are independent with common density function f (t) = e−t, t > 0 . The cost, X, of operating the device until failure is 2T1 + T2 . Which of the following is the density function of X for x > 0 ?

In the solution, the beginning steps are:

P(X<x)=P(2T1+T2<x)

=integral from 0 to x of [integral from 0 to 0.5(x-T2) of (e^-t1)(e^-t2)dt1] dt2

I don't understand where the limits for this integral came from. Specifically the dt2 one, from 0 to x. Could someone please clarify where it came from?

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