# Probability: Sums and Products of Random Variables

• May 11th 2011, 02:44 PM
dizzle1518
Probability: Sums and Products of Random Variables
Suppose that X is uniformly distributed on (0,2), Y is uniformly distributed on (0,3), and X and Y are independent. Determine the distribution functions for the following random variables:

a)X-Y
b)XY
c)X/Y

3. The attempt at a solution

ok so we know the density fx=1/2 and fy=1/3. Since they are independent then fxy(xy)=(1/2)*1/3)=1/6. So we will integrate 1/6 over the rectangle with y (height) 3 and x (width) 2. for a) we have P(X-Y<=Z) = P(X-Z<=Y). This is were I get stuck. I am not sure what my limits of integration are. I know that X-Y can be no less than -3 (0-3). and no more than 2 (2-0). How do I go about finding the limits of integration? Same thing with b and c. For example b yield P(XY<=Z) which is equal to P(Z/X<=Y). So that would be the area of the rectangle under the hyperbola Z/X.
• May 11th 2011, 07:08 PM
mathprofessor
Hi,
I haven't looked at this stuff for decades but instead of looking for P(X-Y<=Z) shouldn't you be trying to find P(x-y = k)???.
Steven
• May 12th 2011, 01:37 AM
Moo
Quote:

Originally Posted by mathprofessor
Hi,
I haven't looked at this stuff for decades but instead of looking for P(X-Y<=Z) shouldn't you be trying to find P(x-y = k)???.
Steven

Easy, it's 0.
They're uniform continuous distributions !
• May 12th 2011, 04:44 AM
mathprofessor
yeah, yeah, you got me. good job!
Steven