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Let $N_t$ a renewal process and $U(t)=E(N_t)$ Let $B_t=S_{N_t}-t$ with $S_{N_t}=X_1+\cdots+X_{N_t}$

Show $\displaystyle U(t)=\frac{t} {E(X_1)}+\frac{E(B_t)}{E(X_1)}$.

I tried: $$\frac{t} {E(X_1)}+\frac{E(B_t)}{E(X_1)}=\frac{E(S_{N_t})}{E(X_1)}=E(X_2+\cdots+X_{N_t})=E(S_{N_t-1})$$ but I don't have $U(t)$

Can you help me?

Thank you

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  • $\begingroup$ I have the solution with the Wald inequality $\endgroup$ – user314729 Feb 15 '16 at 20:14
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    $\begingroup$ So, if you have a solution, why are you asking for? $\endgroup$ – sinbadh Feb 15 '16 at 20:18

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