A machine consists of two components, whose lifetimes have the joint density function

[[math]] f(x,y) = \begin{cases} \frac{1}{50}, \,\, x\gt0, y \gt 0, x+y \lt 10 \\ 0, \, \textrm{Otherwise.} \end{cases} [[/math]]

The machine operates until both components fail. Calculate the expected operational time of the machine.

• 1.7
• 2.5
• 3.3
• 5.0
• 6.7

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: 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.

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.