1 The time to failure, t, in hours, of a machine is often exponentially distributed with a probability density function f(t)=ke-kt, 0≤t<∞o, where k = — and a is 1 a the average amount of time that will pass before a failure occurs. Suppose that the average amount of time that will pass before a failure occurs is 80 hr. What is the probability that a failure will occur in 48 hr or less? The probability is (Round to four decimal places as needed.)

Do you dislike waiting in line? A supermarket chain has used computer simulation and information technology to reduce the average waiting time for customers at 2,300 stores. Using a new system, which allows the supermarket to better predict when shoppers will be checking out, the company was able to decrease average customer waiting time to just 28 seconds.   (a)Assume that supermarket waiting times are exponentially distributed. Show the probability density function of waiting time at the supermarket. f(x) =                                               x ≥ 0         elsewhere (b) What is the probability that a customer will have to wait between 30 and 45 seconds? (Round your answer to four decimal places.)   (c) What is the probability that a customer will have to wait more than 2 minutes? (Round your answer to four decimal places.)

(6) The life in years of a certain type of electronic switch, X, has the following probability density function: f(x) = -=-e-2/², x > 0. (a) Find the value of the constant c. And compute the expected value of x. (b) Find the probability that a randomly selected switch will last more than 3 years. In other words, find P(X > 3). (c) Given that a switch has been working for 1 year already, find the probability that the switch will last more than 3 additional years. (d) If 5 of the new switches are installed in different independent systems, what is the probability that there will be 3 of these 5 switches fail during the first three years after the installation?