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# Homework 3 - 550.430 Introduction to Statistics Spring 2012...

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550.430 Introduction to Statistics Spring 2012 Naiman Homework #3 Due Monday Feb 20th (1) Consider simple random sampling (without replacement) using a sample of size n from a population of size N where each individual i has two attributes x i and y i . Let σ XY = 1 N N i =1 ( x i - μ x )( y i - μ y ) denote the population covariance between x and y where μ x and μ y denote the population means. Let ( X i , Y i ) , i = 1 , . . . , n denote the ( x, y ) values for the individuals sampled. The purpose of this exercise is to derive the covariance between X i and Y j . Assume that the values taken on by the x ’s is ζ 1 , . . . , ζ u and by the y ’s is ω 1 , . . . , ω v and that the number of ( x, y ) pairs taking the value ( ζ i , ω j ) is n ij . We can think of the population is broken into groups labeled 1 , . . . , u according to the x value and into groups labeled 1 , . . . , v according to the y value. We can also think of grouping into u × v groups according to both citeria. (a) If we draw an individual from the population with values X and Y, let I denote the index telling us what x group this person is in (so that X = ζ I ) and let J denote the index telling us what y group this person is in. Thus ( X, Y ) = ( ζ I , ω J ) . Write down the joint probability mass function for ( I, J ) . (b) Suppose we draw two individuals at random without replacement. Let ( I, J ) de- note the index pair for the first individual drawn, and let ( I , J ) denote the index pair for the second individual. Write down the joint probability mass function for

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Homework 3 - 550.430 Introduction to Statistics Spring 2012...

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