4. Barry has a torch that requires two batteries to be operational. In addition to the...

 

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4. Barry has a torch that requires two batteries to be operational. In addition to the empty torch, he has a set of 12 new batteries labelled 1,2,... Barry puts two batteries into the torch and it starts working. Whenever a battery fails the torch stops working and Barry removes the failed one. He then puts in the next unused battery, with the remained working battery, so that the torch works again. Assume that the lifetimes of the batteries are independent random variables, each with an exponential distribution having a mean of 4 months. Let the random variable T be the time the torch fails because the supply of batteries has run out. At that time, exactly one of the original batteries will be working, which we denote by N (also a random variable). (a) What is the expected value of T? (b) What is P(N = 12)? (c) What is P(N = 1)? %3D To get all the marks you must explain your reasoning for each answer. [7 marks] 4. Barry has a torch that requires two batteries to be operational. In addition to the empty torch, he has a set of 12 new batteries labelled 1,2,... Barry puts two batteries into the torch and it starts working. Whenever a battery fails the torch stops working and Barry removes the failed one. He then puts in the next unused battery, with the remained working battery, so that the torch works again. Assume that the lifetimes of the batteries are independent random variables, each with an exponential distribution having a mean of 4 months. Let the random variable T be the time the torch fails because the supply of batteries has run out. At that time, exactly one of the original batteries will be working, which we denote by N (also a random variable). (a) What is the expected value of T? (b) What is P(N = 12)? (c) What is P(N = 1)? %3D To get all the marks you must explain your reasoning for each answer. [7 marks]