Gasoline is to be stocked in a bulk tank once at the beginning of each week and

Question

 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. 

Explanation / Answer

ans=

First, I will verify f(x,y) = 3x for 0 <= y <= x <= 1, 0 for elsewhere

is a valid PDF by using double integration.



(3x) dy dx
lower y = 0
upper y = 1 -x
lower x = 0
upper x =1

Use the change of variable
u = x -y
v = y
The jacobian is 1

Function 3(x) becomes 3 u +3v
Second verify


( 3u+3v) dv du = 1

Lower v =0
Upper v =1 -u
Lower u =0
Lower u =1

So to find PDF of u just integrate with v

answer = ( 3 - 3 u^2) / 2 , 0 <u <1

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