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

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Question: The length of time that a machine operates without failure is denoted by Y and the length of time to repair a failure is

The length of time that a machine operates without failure is denoted

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

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