One hundred items are simultaneously put on a life test. Suppose the life times of the individual items are independent exponential random variables with mean 200 hours. The test will end when there have been a total of 5 failures. If T is the time at which the test ends, find E(T).
Solution: Let $T_i$ be the time between the $i-1$th and $i$th failures. $E(T) = \sum^5_{i=1} T_i = \sum^5_{i=1} \frac{200}{101-i}$. I don't understand how to arrive at this solution. Can someone explain?
Look at a moment when the test begins, or when an item has just died. Suppose that at that moment $k$ items are left alive.
We find the distribution of the minimum additional lifetime among the $k$ survivors. Call this $Y_k$. We use the memorylessness of the exponential. The probability that $Y_k \le y$ is $1$ minus the probability all $k$ items are still alive at time $y$ minutes afterwards. Since the items do not "remember" how long they have been alive, it follows that $\Pr(Y_k\le y)=1-(e^{-1/200})^k$.
Thus $Y_k$ is exponentially distributed with mean $\frac{200}{k}$.
Back to the $T_i$. Since $T=T_1+T_2+\cdots +T_5$, by the linearity of expectation we have $E(T)=E(T_1)+E(T_2)+\cdots+E(T_5)$.
By our earlier calculation, $E(T_1)=\frac{200}{100}$, $E(T_2)=\frac{200}{99}$, and so on up to $E(T_5)=\frac{200}{96}$.
