Suppose the $j^{th}$ light bulb burns for an amount of time $X_j$ and then remains burned out for time $Y_j$ before being replaced. Suppose the $X_j$,$Y_j$ are positive and independent with the $X_j$’s having distribution $F$ and the $Y_j$ ’s having distribution $G$, both of which have finite mean. Let $R_t$ be the amount of time in $[0, t]$ that we have a working light bulb. Show that $\frac{R_t}{t}$ → $\frac{EX_1}{(EX_1 + EY_1)}$ almost surely.
I have shown that $\frac{R_t}{t}$ is in between by $\frac{\sum_{j=1}^{N_t} X_j}{N_t}$ and $1- \frac{\sum_{j=1}^{N_t} Y_j}{N_t}$ ; and both these bounds converge almost surely to what we want to prove in the problem. But how do we conclude that $\frac{R_t}{t}$ also does the same?
