Let $X_1, X_2, \ldots$ be interarrival times like the time between two buses coming to a bus station. Let $Y(t)$ be the time until the next renewal, like the time until the next bus from a given time $t$. This is known as excess. Let $U(t)$ be the time since the last renewal, like the time since the last bus. This is known as the age.
Define the lifetime to be $X(t) = Y(t) + U(t)$. Also known as spread.
Find the value of $E(X(t))$ by conditioning.
How can I do this? I think you need to condition on the first renewal time. I think the answer is $E(X^2)/E(X)$ but I want to understand how to get it with conditioning expectation.
I know $E(Y(t)) = E(X^2)/2E(X)$ if it helps.
Thank you