ECE 434: Random Processes Quiz - Fall 2005, Quizzes of Electrical and Electronics Engineering

A probability quiz from the university of illinois at urbana-champaign's ece 434: random processes course, held in fall 2005. The quiz covers topics such as finding cumulative distribution functions, calculating expected values, and determining independence and conditional densities of random variables.

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Pre 2010

Uploaded on 02/24/2010

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University of Illinois at Urbana-Champaign
ECE 434: Random Processes
Fall 2005
Probability Quiz
Monday, September 12, 2005
Name:
You have one hour for this quiz. The quiz is closed book and closed note.
Calculators, laptop computers, Palm Pilots, two-way e-mail pagers, etc. may not be used.
Write your answers in the spaces provided.
Please show all of your work. Answers without appropriate justification will receive
very little credit. If you need extra space, use the back of the previous page.
Score:
1. (6 pts.)
2. (12 pts.)
3. (8 pts.)
Total: (26 pts.)
1
pf3
pf4

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University of Illinois at Urbana-Champaign

ECE 434: Random Processes

Fall 2005 Probability Quiz

Monday, September 12, 2005

Name:

  • You have one hour for this quiz. The quiz is closed book and closed note.
  • Calculators, laptop computers, Palm Pilots, two-way e-mail pagers, etc. may not be used.
  • Write your answers in the spaces provided.
  • Please show all of your work. Answers without appropriate justification will receive very little credit. If you need extra space, use the back of the previous page.

Score:

  1. (6 pts.)
  2. (12 pts.)
  3. (8 pts.)

Total: (26 pts.)

Problem 1 (6 points) Let X have the pdf fX (x) =

{ (^) sin(x) 2 x^ ∈^ [0, π] 0 else (a) Find the cumulative distribution function FX. In particular, what is FX (2π)?

FX (2π) =

(b) Compute E[sin(X)].

Problem 3 (8 points) Suppose a N (0, 9) random variable X is passed through the quantizer function f shown. The output is Y = f (X).

! 1 ! 2 1 2

1

!0.

! 1

Y=f(x)

x

(a) Express the pmf of Y in terms of the Q function. (Check your answer. Make sure your answer is positive for the right values of y.)

(b) Express the variance of Y in terms of the Q function. Give as simple an answer as possible.