Probability: Sums and Products of Random Variables

  1. 1. The problem statement, all variables and given/known data
    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.
     
  2. jcsd
  3. micromass

    micromass 18,664
    Staff Emeritus
    Science Advisor
    Education Advisor

    Your first integral will be [tex]\int_{\{x-y\leq z\}}{f_X(x)f_Y(y)dxdy}[/tex]. Now, try to determine what values x and y can take on (hint: draw a picture!)
     
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