If X is the amount of money (in dollars) that the salesperson spends on gasoline during the day and Y is the corresponding amount of money (in dollars) for which he or she is reimbursed, the joint density of these two random variables is given by

f(x,y) = { 1/25((20 - x) / x) for 10 < x < 20, x/2 <y < x

0 elsewhere.

Find a) the marginal density of X, b) the conditional density of Y given X = 12, and c) the probability that the salesperson will be reimbursed at least $8 when spending $12.

For a), I tried to find the integral from x/2 to x of f(x,y), but I ended up with [1/25(lnx - x)] evaluated from x/2 to x. There is a mistake there somewhere because I know the correct answer is supposed to be g(x) = (20 - x)/ 50 for 10<x <20.

For b), I used what I know the correct answer for a) was supposed to be and solved f(x,y) / g(x). I got 1/6 and I this is the correct answer.

For c), I tried to use the integral from 8 to 12 of f(x,y), but, again, I am not getting the right answer (which is 1/3).

Any help would be greatly appreciated!