# renewal function and solution(exercice)

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

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