138. A machine consists of two components, whose lifetimes have the joint density
function f(x,y)=1/50 for x>0, y>0, x+y<10
0 otherwise.
The machine operates until both components fail.
Calculate the expected operational time of the machine.
(A) 1.7
(B) 2.5
(C) 3.3
(D) 5.0
(E) 6.7
Below is the solution from the solution manual.
Key: D
Suppose the component represented by the random variable X fails last. This is
represented by the triangle with vertices at (0, 0), (10, 0) and (5, 5). Because the density
is uniform over this region, the mean value of X and thus the expected operational time of
the machine is 5. By symmetry, if the component represented by the random variable Y
fails last, the expected operational time of the machine is also 5. Thus, the unconditional
expected operational time of the machine must be 5 as well.
let T denote the future life time of the machine,
T= max(x,y) where 0<= T <=10
the distribution fuction of T is
F(t) =P(T<t)
=P(max(x,y)<t)
=P(x<t,y<t,x+y<10)
= t*t/50 when 0 < t < 5
1-(10-t)(10-t)/50 when 5 < t < 10
hence ,the expected operational time of the machine E(T) can be solved by the fomula
E(T)=E(1-F(t))
= 5-5/6+5/6 (integating in the interval of (0,5) (5,10) respectively)
=5
as required .
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