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ECE 434 Random Processes Exam, Fall 1999, Exams of Electrical and Electronics Engineering

University of Illinois - Urbana-ChampaignElectrical and Electronics Engineering

Information about an exam for the ece 434 random processes course, which was offered in the fall of 1999. The exam consisted of three problems, worth a total of 25 points, and had a closed-book policy with the exception of one page of notes. The problems involved topics such as upper bounds for random variables, convergence of sequences, and minimum mean square error estimation.

Typology: Exams

Pre 2010

Uploaded on 02/24/2010

koofers-user-ykh
koofers-user-ykh 🇺🇸

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ECE 434 -- RANDOM PROCESSES FALL
1999 Exam I
October 18,1999
You have 75 minutes to complete the exam. The exam will be closed book, except you
may consult one page of notes.
You must show your work for full credit.
There are three problems on three pages, for a total of 25 points.
1. (6 points) Suppose X is a random variable with E[X4]=30.
(a) Derive an upper bound on P[|X|≥10]. Show your work.
(b) (Your bound in (a) must be the best possible in order to get this part correct).
Find a distribution for X such that so that the bound you found in part (a) holds with
equality.
pf3

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ECE 434 -- RANDOM PROCESSES FALL

1999 Exam I

October 18,

You have 75 minutes to complete the exam. The exam will be closed book, except you may consult one page of notes. You must show your work for full credit. There are three problems on three pages, for a total of 25 points.

  1. (6 points) Suppose X is a random variable with E[X^4 ]=30. (a) Derive an upper bound on P[|X|≥10]. Show your work. (b) (Your bound in (a) must be the best possible in order to get this part correct). Find a distribution for X such that so that the bound you found in part (a) holds with equality.
  1. (9 points) Let Q be uniformly distributed on the interval [0,2p] and let Xn=cos(n). (Hint: cos(nq)cos(mq)=(cos((n-m) q)+cos((n+m) q))/2. (a) Does Xn converge in almost surely as n tends to infinity? Justify your answer. (b) Does Xn converge in mean square sense as n tends to infinity? Justify your answer. (c) Does Xn converge in distribution as n tends to infinity? Justify your answer.