06_spring_exam1

06_spring_exam1 - Midterm Exam I ECE534 Spring 2006 There...

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Unformatted text preview: Midterm Exam I ECE534 Spring 2006 There are a total of five problems Mar. 8, 7:00-8:30 pm You are allowed one sheet (two pages) of notes; no calculators. Each problem is worth 20 points Please put your NAME here: 1. (a) Let X be a uniformly distributed random variable on [0 , 1] . Find the characteristic function of X. (b) Let Y 1 , Y 2 ,... be a sequence of independent random variables uniformly distributed over { , 1 , 2 ,..., 9 } . Find the characteristic function of Y i . (c) Let X n = ∑ n i =1 Y i 10- i . Find the characteristic function of X n . (d) Does the sequence X 1 , X 2 , X 3 ,... converge in distribution? If so, what is the limiting distribution? Clearly justify your answer. (e) Does the sequence X 1 , X 2 , X 3 ,... converge almost surely. If so, what is the distribution of the random variable to which it converges? Clearly justify your answer. (Hint: The following fact may be useful: a sequence of non-decreasing, upper-bounded real numbers has a finite limit.) 1 2. Suppose you toss a fair toss coin2....
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This note was uploaded on 01/05/2010 for the course ECE 01 taught by Professor All during the Fall '09 term at Aarhus Universitet.

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06_spring_exam1 - Midterm Exam I ECE534 Spring 2006 There...

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