Chapter 12, Problem 18RE

Deciding on a Service Contract The condenser motor in an air conditioner costs $\$300$ to replace, but a home air-conditioning service will guarantee to replace it free when it burns out if you will pay an annual insurance premium of $\$25$. The life span of the motor is an exponential random variable with an expected life of $10$ years. Should you take out the insurance for the first year? [Hint: Consider the random variable $Y$ such that $Y=300$ if the motor burns out during the year and $Y=0$ otherwise. Compare $E\left(Y\right)$ with the cost of $1$ year's insurance.]