The value of a piece of factory equipment after three years of use is 100(0.5)[math]X[/math] where [math]X[/math] is a random variable having moment generating function

[[math]] M_X(t) = \frac{1}{1-2t}, \, t \lt \frac{1}{2}. [[/math]]

Calculate the expected value of this piece of equipment after three years of use.

• 12.5
• 25.0
• 41.9
• 70.7
• 83.8

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

Solution: C

[[math]] \begin{align*} E[100(0.5)^X] = 100 E[(0.5)^X] = 100E[e^{(\ln(0.5)}X] &= 100M_X(\ln(0.5))\\ &= 100 \frac{1}{1-2\ln(0.5)}\\ &= 41.9. \end{align*} [[/math]]