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Unformatted text preview: X 1 ,X 2 ,... convergence in probability to a constant μ iﬀ the sequence also converges in distribution to μ . Central Limit Theorem Let X 1 ,X 2 ... with existing MGF, with EX = μ and V ar ( X ) = σ 2 > 0. Let G n ( x ) denote the cdf of Z = √ n ( ¯ X n-μ ) σ . Then lim n →∞ G n ( x ) = Z x-∞ 1 √ 2 π e-y 2 / 2 dy, in other words Z has limiting N (0 , 1) distribution. 1 The CLT is actually also true for ANY distribution with EX = μ and 0 < V ar ( X ) < ∞ . Example 5.5.16: Normal approx to NegBin Slutsky’s Theorem 5.5.17 If X n → X in distribution and Y n → a , a constant, in probability, then • Y n X n → aX in distribution • X n + Y n → X + a in distribution Example 5.5.18: Normal approximation with estimated variance 2...
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