EECS 501__________________________PROBLEM SET #4__________________________Fall 2001

ASSIGNED: Sep. 28, 2001. Read Stark and Woods pp. 119-178. THIS WEEK:
DUE DATE: Oct. 05, 2001. Joint, conditional, and transformed pdfs and PDFs.
  1. Stark and Woods #3.23. A 2-D transformation of joint RVs.
  2. Stark and Woods #3.24. Choosing a linear transformation to decorrelate two RVs.
    Use a=b=c=1 and d=-1. This will be much easier after studying covariance matrices.
  3. Two points x and y are chosen at random from the interval [0,1]. x and y are independent.
    Compute the pdf fd(D) where d=|x-y| is the distance between x and y.
  4. RVs x and y have joint pdf fx,y(X,Y)=AX if 1 < X < Y < 2; fx,y(X,Y)=0 otherwise.
    1. Compute the constant A.
    2. Compute the marginal pdf fy(Y).
    3. Compute the conditional pdf fx|y(X|Y=3/2).
    4. Compute the pdf fz(Z) for random variable z=y-x.

  5. RVs x and y have joint pdf fx,y(X,Y)=CXY2 if 0 < |Y| < X < 1; fx,y(X,Y)=0 otherwise.
      C is a constant; A is the event A={(X,Y):XY > 1/4}, which is bordered by a hyperbola.
    1. Compute the constant C.
    2. Compute Pr[A].
    3. Compute the conditional marginal pdf fx|A(X|A).
    4. Compute the PDF Fw(W) and pdf fw(W) for w=loge(|y|)/loge(x).
      WARNING: Watch your signs on part (d).

    Excuse heard in a genetic engineering class: "My homework ate the dog."