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University of Illinois at Urbana-Champaign
ECE 534: RANDOM PROCESSES
Fall 2007
Probability Quiz
Monday, September 10, 2007
Name:
- This is a closed-book exam. No notes of any kind are allowed.
- Calculators, laptop computers, Palm Pilots, two-way email pagers, etc. may not be used.
- Write your answers in the space provided. If you need extra space, use the back of the previous page.
- Please show all of your work. Answers without appropriate justification will receive very little credit.
Score: 1: (12 points) 2: (12 points) 3: (12 points) Total: (36 points)
1 Suppose continuous RVs X and Y have the joint pdf
fX,Y (x, y) =
{ (^) x C if 1^ ≤^ x^ ≤^ 3 and 0^ ≤^ y^ ≤^2 0 otherwise
Let R be defined by R = Y /X and let D be the event R ≤ 1 /3. a) Determine the value of C.
b) Are X and Y independent? Are they independent conditioned on D?
3 Suppose the professional sports playoffs have begun and we have m games in the first round (2m teams). Because of new stringent testing procedures, each team has been dis- qualified due to excessive player steroid use, independently, with probability p. Let N be the number of teams still eligible after the disqualifications. Let C be the number of games that will be played in the first round (i.e., the number of games in which both teams are still eligible). Find E[C|N = n]. (hint: remember that the expectation is a linear operator).
Extra space if necessary. Please denote which problem you are using this extra space for.
