Gasoline is to be stocked in a bulk tank once at the beginning of
each week and then sold the individual customers. Let X denote the
proportion of the capacity of the bulk tank that is stocked at the
start of the week. Let Y denote the proportion of the capacity of
the bulk tank that is sold during the week. Because X and Y are both
proportions, both variables take on values between 0 and 1. Further,
the amount sold, Y, cannot exceed the amount available, X. Suppose
that a joint pdf model for X and Y is given by:
f(x,y) = 3x for 0 <= y <= x <= 1, 0 for elsewhere
Find the pdf for U = X - Y, the proportional amount of gasoline
remaining at the end of the week. Use the pdf of U to find the
expected value of U, E(U). Use the pdf of U to find the probability
that the proportional amount of gasoline remaining at the end of
the week, U, is between 0.25 and 0.5.
If anyone can help me get the problem started or guide me in the right direction, it would be greatly appreciated. I'm not sure how to start this problem.