Question

The value of a piece of factory equipment after 3 years of use is V=100(0.5)^x where X is a random variable having moment generating function M_X(t)=(1-2t)^-1,t<1/2. Calculate the expected value of this piece of equipment after three years of use.

Answer 1

By the problem the value of a piece of factory equipment after three years of use is $V=100(0.5)^X$ where X is a random variable having moment generating function

$M_X(t)=E(e^{tX})=\frac{1}{(1-2t)}, for t<\frac{1}{2}.$

So the expected value of this piece of equipment after three years of use

$E(V) = E(100(0.5)^X) = 100E(0.5^X) \\ E(V) = 100E(e^{ln0.5^X})=100E(e^{Xln0.5})=100M_X(ln0.5) \\ ln(0.5)= -0.6931 <0.5 \\ E(V) = \frac{100}{1-2 \times (-0.6931)}= 41.91$