Mr. Smith immediately replaced the battery on his radio after the radio dead / did not work. Suppose the time required to replace the battery is neglected because the time is very small when compared to the life of the battery. Let N(t) represent the number of batteries that have been replaced during the first t years of the radio's life, without counting the batteries used when the radio was started.

a. Suppose that battery life is a random event that has an identical and independent distribution. What is the N(t) renewal process? Explain your answer.

b. If the battery life is a random variable whose iid (independent and identically distribution) follows a uniform distribution at intervals of (1, 5) years. Determine the battery replacement rate in the long term

c. If Mr. Smith decided to keep replacing the battery if it had reached 3 years of use even though the battery was still functioning. The cost to replace the battery is $75 if replacement is planned (ie up to 3 years of use), and $125 if the battery is malfunctioning/damaged. Suppose C(t) represents the total cost incurred by Mr. Smith up to time t. Is the C(t) renewal reward process? Explain your answer.

d. find the average cost incurred by Mr. Smith in 1 year.