EECS 501__________________________PROBLEM SET #4__________________________Fall 2001

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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.

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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.

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3. Two points x and y are chosen at random from the interval [0,1]. x and y are independent.
   Compute the pdf f_d(D) where d=|x-y| is the distance between x and y.

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4. RVs x and y have joint pdf f_x,y(X,Y)=AX if 1 < X < Y < 2; f_x,y(X,Y)=0 otherwise.
   1. Compute the constant A.
   2. Compute the marginal pdf f_y(Y).
   3. Compute the conditional pdf f_x|y(X|Y=3/2).
   4. Compute the pdf f_z(Z) for random variable z=y-x.

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5. RVs x and y have joint pdf f_x,y(X,Y)=CXY² if 0 < |Y| < X < 1; f_x,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 f_x|A(X|A).
   4. Compute the PDF F_w(W) and pdf f_w(W) for w=log_e(|y|)/log_e(x).
      WARNING: Watch your signs on part (d).

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