# Mean excess life conditionally on current life in a renewal process

I am self-studying renewal processes and I came across an interesting problem. How would one go about finding the expectation of excess life given that current life is equal to x? I am assuming in this problem that the renewal process has an interarrival distribution function of F(x).

i.e. my problem is: What is E[excess life | current life = x]?

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$$\frac1{1-F(x)}\int_0^{+\infty}tf(x+t)\mathrm dt.$$ A refined result is that, calling $C_y$ and $L_y$ the current life and the excess life at time $y$, for every $0\leqslant x\leqslant y$ the distribution of $L_y$ conditionally on $C_y=x$ is independent on $y$ and has density $g_x$, where $$g_x(t)=\frac{f(x+t)\cdot[t\gt0]}{1-F(x)}.$$