In this problem, we will modify the light bulb problem of Exercise 2.74. Recall that there were two types of light bulbs, long-life (L-type) and short-life (S-type) and we were given an unmarked bulb and needed to identify which type of bulb it was by observing how long it functioned before it burned out. We know that the apriori probabilities are Pr(S)=0.75 and Pr(L)=0.25. First the two conditional distributions of the bulb lifetimes will be modified so that they are continuous random varialbes with exponential distributions. That is, fXS(x)=1001exp(100x)u(x) and fXL(x)=10001exp(1000x)u(x) where X is the random variable that measures the lifetime of the bulb in hours. Second, the observation of the experiment will be modified as follows. The light bulb to be tested will be turned on at 5pm on Friday and will be allowed to burn all weekend. We will come back and check it on Monday morning at 8am and at that point it will still be lit or it will have burned out. Note that since there are a total of 63 hours between the time we start and end the experiment and we will not be watching the bulb at any time in between, there are only two possible observations in this experiment, { the bulb burnt out }{X63} (a) Given the bulb burnt out over the weekend, what is the probability that the bulb was S-type? (b) Given it is observed that the bulb tested is still lit at the end of the weekend, what is the probability that the bulb was L-type? (c) Suppose we decide the bulb was S-type if we observe {X63}. What is the probability that our decision is wrong? In this problem, we will modify the light bulb problem of Exercise 2.74. Recall that there were two types of light bulbs, long-life (L-type) and short-life (S-type) and we were given an unmarked bulb and needed to identify which type of bulb it was by observing how long it functioned before it burned out. We know that the apriori probabilities are Pr(S)=0.75 and Pr(L)=0.25. First the two conditional distributions of the bulb lifetimes will be modified so that they are continuous random varialbes with exponential distributions. That is, fXS(x)=1001exp(100x)u(x) and fXL(x)=10001exp(1000x)u(x) where X is the random variable that measures the lifetime of the bulb in hours. Second, the observation of the experiment will be modified as follows. The light bulb to be tested will be turned on at 5pm on Friday and will be allowed to burn all weekend. We will come back and check it on Monday morning at 8am and at that point it will still be lit or it will have burned out. Note that since there are a total of 63 hours between the time we start and end the experiment and we will not be watching the bulb at any time in between, there are only two possible observations in this experiment, { the bulb burnt out }{X63} (a) Given the bulb burnt out over the weekend, what is the probability that the bulb was S-type? (b) Given it is observed that the bulb tested is still lit at the end of the weekend, what is the probability that the bulb was L-type? (c) Suppose we decide the bulb was S-type if we observe {X63}. What is the probability that our decision is wrong