# TA6 - ECO2121 Methods of Economic Statistics TA6 25...

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ECO2121: Methods of Economic Statistics TA6 25 February 2009 A statistician is like an ice-cream chef. 1. Normal Distribution (a) Suppose X ~ Normal( μ , σ 2 ), then a X +b ~ Normal(a μ +b, a 2 σ 2 ). Hence, we get the standardization : Z = ( X μ )/ σ ~ Normal(0, 1) (b) Suppose X ~ Normal( μ X , σ X 2 ) and Y ~ Normal( μ Y , σ Y 2 ) independently , then X ± Y ~ Normal( μ X ± μ Y , σ X 2 + σ Y 2 ). (c) This rule could be generalized as follows: The distribution of the linear combination of normally distributed random variables is Normal. Hence we need only the mean and the variance of the linear combination to characterize itself. By (revisited TA3 and TA5) E [ aX + bY ] = a E X + b E Y Var[ aX ± bY ] = a 2 Var[ X ] + b 2 Var[ Y ] if X and Y are independent We could know the target distribution. So following (b), what is the distribution of aX + bY ? 2. Convergence in Distribution (Central Limit Theorem) For a random variable X of any distribution such that E X = μ and Var[ X ] = σ 2 exist and are finite, and X 1 , , X n are an i.i.d. sample, as n →∞

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