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Unformatted text preview: Stat 332 R.J. MacKay, University of Waterloo, 2005 Chapter 8 Solutions 1 Chapter 8 Exercise Solutions 1. Find the quadratic expansion of f x y y x ( , ) / = about the point ( ( ), ( )) μ μ x y to estimate the bias in the estimator ~ ~ ( ) / ~ ( ) θ μ μ = y x . Note that the general form of the expansion is f x y f x y f x y x x x f x y y y y f x y x x x f x y x y x x y y f x y y y y ( , ) ( , ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( , ) ( )( ) ( , ) ( ) ≈ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ + ∂ ∂ 2 2 2 2 2 2 2 2 2 This quadratic function has the same value, first and second derivatives at the point ( , ) x y as does f x y ( , ). You can easily check this statement by differentiating the right side of the expression. To use the expansion, we have ∂ ∂ =  ∂ ∂ = ∂ ∂ = ∂ ∂ ∂ =  ∂ ∂ = f x y x f y x f x y x f x y x f y μ μ μ μ μ μ ( ) ( ) , ( ) , ( ) ( ) , ( ) , 2 2 2 3 2 2 2 2 1 2 1 0 so we can write ~ ( ) ( ) [ ~ ( ) ( )] ( ) [ ~ ( ) ( )] ( ) ( ) [ ~ ( ) ( )] ( ) [ ~ ( ) ( )][ ~ ( ) ( )] θ θ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ μ ≈  + + y x x x x y y y x x x x x x y y 2 3 2 2 1 2 2 1 and E y x Var x x Cov x y x Var x Cov x y ( ~ ) ( ) ( ) ( ~ ( )) ( ) ( ~ ( ), ~ ( )) ( ) [ ( ~ ( )) ( ~ ( ), ~ ( )] θ θ μ μ μ μ μ μ θ μ θ μ μ μ ≈ + = + 3 2 2 1 1 The approximate bias is given by the second term. We know Var x f x n ( ~ ( )) ( ) ( ) μ σ = 1 2 and with a bit of effort, we can show Cov x y f Cov x y n ( ~ ( ), ~ ( )) ( ) ( , ) μ μ = 1 where Cov x y ( , ) is the population covariance. The key point is to notice that the bias has a factor 1 n and will be small if the sample size is large. 2. In order to count the number of small items in a large container, a shipping company selects a sample of 25 items and weighs them. They then weigh the whole shipment (excluding the container). Assume that there is small error in weighing and act as if SRS is used  it is not, the sampling is haphazard. Let the weight of the ith item in the population be y i and the total known weight be τ a) Show that an estimate of the population size is $ / N y i i s = ∑ τ ε 25 . Stat 332 R.J. MacKay, University of Waterloo, 2005 Chapter 8 Solutions 2...
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 Spring '09
 Xu(Sunny)Wang
 Statistics, Normal Distribution, Standard Deviation

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