How do I use expected value in this question?

A car comes equipped with one spare tire. The lifetimes of the four tires at the beginning of a long distance journey are independent identically distributed exponential random variables with a mean of 5000 miles. The spare tire has an exponential lifetime with a mean of 1000 miles. Compute the expected number of miles that can be covered without having to go to a tire shop. Note that when one of the regular tires fails, the driver will replace it with the spare one and then when one of the remaining three regular tires or the spare tire fails, he will go to the body shop.

My current logic is to find the expectation of the minimum of the 4 tires and then the expectation of the minimum of 3 tires and a spare tire. Add those two other and thats my answer. Can someone confirm if this is the right method?

Solution 1:

Your approach is fine.

Guide:

Key steps that you need:

• The minimum of independent expoenential distribution remains an exponential distribution, compute the expected value.

• Use the memoryless propery of exponential distribution.

Compute the following:

$$E(\min(X_1, X_2, X_3, X_4)) + E(\min(Y, X_1, X_2, X_3)) $$

where $X_i \sim \exp\left(\frac1{5000}\right), Y\sim\exp\left(\frac1{1000} \right).$