The length of time that a machine operates without failure is denoted by X 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. X and Y are independent and each has the density function f(x)= e^-x, x>0 , 0 elsewhere f(y) = e^-y, y>0 , 0 elsewhere
Find the probability density function for U = X/(X+Y), the proportion of time that the machine is in operation during any one operation-repair cycle.
I know that f(x,y)= e^-(x+y), x>0,y>0 because X and Y are independent and 0 elsewhere also my U=X/(X+Y) and I can set V to be ether X,Y or X+Y but I don't know which is best to chose.. in order to make this transformation I need to find Jacobian |J| that is far as I get..Can anyone help me with this problem? Thank you!
The choice of the transformation variable should be made in such a way that the transformation is one-to-one so that the new variables can explicitly be written completely in terms of the old variables and vice versa. Now for the Jacobian calculation, other than the choice to be a one-to-one transformation, we also make sure the derivation of partial differentiation can easily be derived. In this case, $(U,V) = (X/(X+Y), X+Y)$ should be the best choice.