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Unformatted text preview: Chapter 6.4 The Sample Variance Let the random sample,, . . . , be a set of identically distributed and independent random variables with mean and variance. 1 X 2 X n X P 2 V The sample mean is defined as: n 1i i X n 1 X Previous work has studied the properties of the sampling distribution of the sample mean. A familiar result is: V P n , N~X 2 Now consider the sample variance defined as the random variable: n 1i 2 i 2 X )X X( 1 n 1 s What are the properties of the sampling distribution of ? 2 X s Econ 325 Chapter 6.4 1 A result is: 5 V 2 X sE This can be shown as follows. ^` >@ >@ 5 5 5 V V 0V P0 P0 P0 P0 P0 0P0 )1 n( n n n ) X(En ) X(E ) X(n ) X( E ) X() X( E )X X( E n 1i 2 2 i n 1i 2 2 i n 1i 2 i n 1i 2 i Therefore, +, >@ 5 5 V V )1 n( 1 n 1 sE 2 X That is, the divisor of (n 1) ensures that gives an unbiased estimation rule for the population parameter 2 X s 2 . This justifies the use of (n 1) in the divisor. This justifies the use of (n 1) in the divisor....
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This note was uploaded on 01/28/2011 for the course ECON 325 taught by Professor Whistler during the Spring '10 term at The University of British Columbia.
- Spring '10