09_09_09_0 - STAT 410 Fal 2009 Examples for 09/09/2009 Let...

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STAT 410 Examples for 09/09/2009 Fal 2009 Let X 1 , X 2 , … , X n be a random sample from a population with the population mean μ and the population standard deviation σ . By CLT, X – μ is approximately N ( 0 , n 2 σ ) for large n . If g ( x ) is differentiable at μ and x is “close” to μ , g ( x ) g ( μ ) + g ' ( μ ) ( x μ ). Therefore, if g ' ( μ ) 0, g ( X ) is approximately N ( g ( μ ) , [ g ' ( μ ) ] 2 n 2 σ ) for large n .
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1. If the random variable Y denotes an individual’s income, Pareto’s law claims that P ( Y y ) = θ y k , where k is the entire population’s minimum income. It follows that f Y ( y ) = 1 θ θ 1 θ + y k , y k ; θ > 1. Assume k is known. Let Y 1 , Y 2 , … , Y n be a random sample of size n . Recall that the method of moments estimator θ ~ of θ , is k - = Y Y θ ~ . Show that
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09_09_09_0 - STAT 410 Fal 2009 Examples for 09/09/2009 Let...

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