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University of Illinois at Urbana-Champaign
ECE 534: RANDOM PROCESSES
Fall 2006
Probability Quiz
Monday, September 11, 2006
Name:
- This is a closed-book exam.
- 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:
- (12 points)
- (12 points)
- (12 points) Total. (36 points)
1 A group of Michigan Alumni at GoBlue Airlines just purchased trou- bled Air Illini. Air Illini currently only offers service to Springfield and Decatur. Because they are so unorganized, flights occur in a ran- dom manner and often get cancelled. The probability that a flight to Springfield or Decatur is cancelled is 1/5 and 1/10 respectively. Any particular flight goes to Springfield with probability 2/3 and Decatur with probability 1/3. a) What is the probability that a flight gets cancelled?
b) What is the expected number of flights before the first cancella- tion?
2 Suppose X 1 ,... , Xn are independent and identically distributed (i.i.d.) random variables with distribution FX (x). Define U and L as
L = min (X 1 ,... , Xn) U = max (X 1 ,... , Xn)
a) Find FU (u). Hint: consider equivalence of events.
b) Find FL (l). Hint: consider 1 − FL (l) and equivalence of events.
c) Are they independent? Justify your answer.
c) Find P (X 1 ≤ X 2 ). Note that minimal computation is required.
