EECS 501__________________________PROBLEM SET #3__________________________Fall 2001

ASSIGNED: Sept. 21, 2001. Read Stark and Woods pp. 60-119 on various pdfs.
DUE DATE: Sept. 28, 2001. THIS WEEK: pdfs and PDFs and sample spaces.
1. Stark and Woods #2.2. A simple PDF and pdf problem. Your sketches should be neat.
2. Prove for n=3 only (not for general n) the principle of inclusion and exclusion:

1. Use this result and some permutations/combinations to solve the following problem:
2. An absent-minded professor (no one you know) wrote n letters and sealed them in
envelopes without writing the addresses on the envelopes. Having forgotten which letter he
had put into each envelope, he wrote the n addresses on the n envelopes entirely at random.
Compute Pr[at least one envelope was addressed correctly].
HINT: Let Ai=Pr[ith envelope addressed correctly]. What happens as n goes to infinity?

3. Random variables x,y,z,w have the joint pdf below. Compute the pdf fx|w,y,z(X|W,Y,Z).

4. Let a,b,c > 0 be independent random variables, each with pdf fx(X) and PDF Fx(X). Show:

5. Two real numbers b and c are chosen at random and independently of each other.
Compute Pr[the quadratic equation t2+bt+c=0 has two real roots] as follows:
Let b and c each be chosen from the interval [-n,n] and solve the problem for some finite n
(draw your picture carefully--it's tricky!). What happens as n goes to infinity?
6. The rejection technique is a procedure for generating random numbers with a desired pdf fx(X):
Let fx(X) be nonzero only for 0 < X < 1, and have maximum value fm. Do the following:
• Spin a wheel of fortune repeatedly; group the resulting numbers into pairs (Xi,Yi)
• Keep Xi only if fx(Xi) > fmYi; otherwise discard it.
The problem for you is to do the following:
1. Show that the accepted Xi have pdf fx(X). (This is a conditional marginal pdf problem.)
2. Write a simple Matlab program that uses the rejection technique to generate
random numbers distributed with the parabolic pdf fx(X)=3X², 0 < X < 1.
Use Matlab's hist to create a histogram (bar graph) of the number of accepted Xi.
Turn in: Matlab code and histogram. Use 20 bins and 4000 random number pairs.

"A chicken is an egg's way of making another egg"--anonymous.