10.4 Exercise 3

Present value of future costs. Suppose money may be invested at an annual rate a, compounded continually. Then one dollar in hand now, has a value $e^{a x}$ at the end of x years. Hence, one dollar spent x years in the future has a present value$e^{- a x}$. Suppose a device put into operation has time to failure (in years) $X \sim$ exponential $\left(\lambda\right)$. If the cost of replacement at failure is C dollars, then the present value of the replacement is $Z = C e^{- a X}$. Suppose $\lambda = 1 / 10$, $a = 0 . 07$, and $C = \$ 1000$.

1. Use the result of the exercise to determine the probability $Z \leq 700 , 500 , 200$.
2. Use a discrete approximation for the exponential density to approximate the probabilities in part (a). Truncate X at 1000 and use 10,000 approximation points.