The length of time that a machine operates without failure is denoted by Y and the...

 

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The length of time that a machine operates without failure is denoted by Y and the length of time to repair a failure is denoted by Y. After a repair is made, the machine is assumed to operate like a new machine. Y and Y2 are independent and each has the same density function JY.(1) = { 1, exp(-1/8), 0 <1 otherwise Let U=Y/(Y+Y), the proportion of time that machine 1 is in operation during any one operation-repair cycle and U = Y + Y2 the length of the cycle. a) Find the joint distribution of U and U2. b) Find the probability density function for U. c) Determine the covariance between U and U2. d) Find the probability P(Y < y U2 = u2) for 0 The length of time that a machine operates without failure is denoted by Y and the length of time to repair a failure is denoted by Y. After a repair is made, the machine is assumed to operate like a new machine. Y and Y2 are independent and each has the same density function JY.(1) = { 1, exp(-1/8), 0 <1 otherwise Let U=Y/(Y+Y), the proportion of time that machine 1 is in operation during any one operation-repair cycle and U = Y + Y2 the length of the cycle. a) Find the joint distribution of U and U2. b) Find the probability density function for U. c) Determine the covariance between U and U2. d) Find the probability P(Y < y U2 = u2) for 0