The length of time that a machine operates without failure

Chapter 6, Problem 98SE

(choose chapter or problem)

The length of time that a machine operates without failure is denoted by \(\mathrm{Y}_{1}\) and the length of time to repair a failure is denoted by \(\mathrm{Y}_{2}\). After a repair is made, the machine is assumed to

operate like a new machine. \(Y_{1} \text { and } Y_{2}\) are independent and each has the density function

\(f(y)=\left\{\begin{array}{ll}  e^{-y}, & y>0 \\  0, & \text { elsewhere. }  \end{array}\right.\)

Find the probability density function for \(U=Y_{1} /\left(Y_{1}+Y_{2}\right)\), the proportion of time that the machine is in operation during any one operation–repair cycle.

Equation Transcription:

 {

Text Transcription:

Y1

Y2

Y1 and Y2

f(y)= {e^-y,     y > 0     0, elsewhere  

U=Y1/Y1+Y2)