91Ó°ÊÓ

Suppose that n items are being tested simultaneously.

The items are independent and length of each item follows exponential distribution with parameter \(\beta \)

The length of time\({Y_1}\)until one component fails has exponential distribution with parameter\(n\beta \):

\(E\left( {{Y_1}} \right) = \frac{1}{{n\beta }}\).

The additional length of time\({Y_2}\)until a second component fails has the exponential distribution with parameter .

\(E\left( {{Y_2}} \right) = \frac{1}{{\left[ {\left( {n - 1} \right)\beta } \right]}}\)

Similarly

\(E\left( {{Y_3}} \right) = \frac{1}{{\left[ {\left( {n - 2} \right)\beta } \right]}}\).

The total time until three components fail is:

\({Y_1} + {Y_2} + {Y_3}\)

And

\(E\left( {{Y_1} + {Y_2} + {Y_3}} \right) = \left( {\frac{1}{n} + \frac{1}{{n - 1}} + \frac{1}{{n - 2}}} \right)\frac{1}{\beta }\).