Determine the seller's payment expectations for the warranty

Question

Suppose the value of an instrument (v) is based on the number of years since purchase (t), thus

$v(t)=e^{7-0.2t}$

If the tool is damaged in the first 7 years since the tool was purchased, then the buyer can make a claim on the guarantee where the seller will pay an amount of money to the buyer according to the value of the tool when the tool is damaged. If after 7 years the tool is damaged then the buyer will not get anything. The lifetime of the tool until it breaks is exponentially distributed with a mean of 10. Determine the seller's payment expectations for the warranty.

The answer is supposed to be 320.78. Kindly help to get that answer :)

Answer 1

Hint:

Since payouts are zero after $7$ years, we can model the payout $C$ as a function of the exponentially distributed failure time $T$ as follows:

$$C = \mathbf{1}_{\leq 7}(T)e^{7-0.2T}$$

The expected value of $C$ is:

$$E[C] = \int_0^{\infty}\left(\mathbf{1}_{\leq 7}(t)e^{7-0.2t}\right)0.1e^{-0.1t}dt = 0.1\int_0^{7}\left(e^{7-0.2t}\right)e^{-0.1t}dt \\= 0.1\int_0^{7}e^{7-0.3t} dt =0.1e^7\int_0^{7}e^{-0.3t} dt $$

Can you solve this definite integral?