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.

- Stark and Woods #3.23. A 2-D transformation of joint RVs.
- 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.

- 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.

- 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.
- Compute the constant A.
- Compute the marginal pdf f
_{y}(Y).
- Compute the conditional pdf f
_{x|y}(X|Y=3/2).
- Compute the pdf f
_{z}(Z) for random variable z=y-x.

- RVs x and y have joint pdf f
_{x,y}(X,Y)=CXY^{2} 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.
- Compute the constant C.
- Compute Pr[A].
- Compute the conditional marginal pdf f
_{x|A}(X|A).
- 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).

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