> 
> One hundred items are simultaneously put on a life test. Suppose the
>  lifetimes 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 $E[T]$.
> 
> 
> 

My attempt:
Let $T\_i$ be the interarrivial time between the $(i-1)$th failure and $i$th failure. Then we have $$T= \sum\_{i=1}^5 T\_i$$ which is quickly followed by $$E[T]= \sum\_{i=1}^5 E[T\_i].$$

I guess there is no problem above. I now claim that interarrivial time is independent exponential r.v.. Thus, I have $E[T\_i] = \frac{1}{200}$, which differs from the book answer: $E[T\_i] = \frac{200}{101-i}$. Can anyone point out my mistake?

Thanks in advance.