stat610hw1 - Statistics 610: Homework 1 September 16, 2008...

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Statistics 610: Homework 1 September 16, 2008 Announcement: The homework carries 140 points but the maximum you can score is 110 points. (1) Recall the Democrat–Republican experiment we discussed in class. There are N individuals in a town marked by numbers 1 , 2 ,...N . Out of these an unknown m are Democrats, and p = m/N is this unknown proportion. We draw n individuals one by one, without replacement. Let X i denote the affiliation of the i ’th voter: 1 if Democrat and 0 if Republican. The X i ’s are Bernoulli random variables. Our interest is in studying the distribution of the X i ’s. To this end, we can think of the experiment of drawing individuals continuing till stage N , so that we extend our sequence of random variables X 1 ,X 2 ,...,X n to X 1 ,X 2 ,...,X n ,X n +1 ,...,X N . The underlying sample space Ω is the set of all permutations of the integers 1 through N and X i ( ω ) is the i ’th element in the permutation ω . All permutations are equally likely. Use this formulation to (a) Work out the distribution of X 1 . (b) Work out the distribution of ( X 1 ,X 2 ) . (c) Argue that all X i ’s have the same distribution. (d) All pairs ( X i ,X j ) have the same distribution, and more generally all k –tuples have the same distribution. Thus, the distributions are invariant to the ”time factor” which is not immediate at first glance.
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This note was uploaded on 04/14/2010 for the course STATS 610 taught by Professor Moulib during the Fall '09 term at University of Michigan.

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stat610hw1 - Statistics 610: Homework 1 September 16, 2008...

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