Question

3. Let X, Y, and Z be independent unit exponential random variables, with common density f(t) = e^(-t) for t > 0.

Let T_1 = min (X, Y, Z )

T_2 = middle value of the three numbers X, Y, Z

T_3 = max (X, Y, Z )

(a) Find P( T_1 > t ) for t >0.

(b) Find P( T_3 < t ) for t > 0.

(c) Find P( T_2 >  t ) for t > 0.

HINT: T_2 > t happens when how many of X and Y and Z are greater than t ?

(d)  Find E ( T_3 - T_2 ) = expected difference between T_3 and T_2 .

HINT: One way to do part (d) is obviously to find the densities of T_2 and T_3 from the answers to parts (b) and (c) and then to use those densities to calculate E( T_2 ) and E(T_3 ). You could also integrate the survival functions (See page 332, under "Expectation from the survival function"). A much easier way is to just write down the answer, which you can do if you use the memoryless property of exponential distributions. Think about 3 light bulbs with independent unit exponential lifetimes. As long as such a bulb is working, its future behavior is exactly the same as the future behavior of a new bulb.

(e) Find E(T_3) and var(T_3).

HINT: There is almost no work involved in doing part (e) if you figured out the clever way to do part (d) and you use the equality

T_3 = T_1 + (T_2 - T_1) + (T_3 - T_2 ).

Answer 1