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Unformatted text preview: 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 XY = 1 N N i =1 ( x i- x )( y i- y ) denote the population covariance between x and y where x and 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 1 , . . . , u and by the y s is 1 , . . . , v and that the number of ( x, y ) pairs taking the value ( i , 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 = I ) and let J denote the index telling us what y group this person is in. Thus ( X, Y ) = ( I , 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...
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This note was uploaded on 02/26/2012 for the course STATISTICS 530 taught by Professor Naiman during the Fall '11 term at Johns Hopkins.
- Fall '11