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# 10_19 - STAT 410 Examples for Fall 2011 Let X n be a...

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Unformatted text preview: STAT 410 Examples for 10/19/2011 Fall 2011 Let { X n } be a sequence of random variables and let X be a random variable. Let F X n and F X be, respectively, the c.d.f.s of X n and X. Let C ( F X ) denote the set of all points where F X is continuous. We say that X n converges in distribution to X if ( ) ( ) x x n n X X F F lim = ∞ → , for all x ∈ C ( F X ). We denote this convergence by X X D n → . Let X 1 , X 2 , … be an infinite sequence of random variables, and let X be another random variable. Then the sequence { X n } converges in probability to X, if for all ε > 0, ( ) X X P lim ε = ≥- ∞ → n n , ( ) 1 X X P lim ε = <- ∞ → n n , and write X X P n → . Theorem 1 X X P n → ⇒ X X D n → Theorem 2 b D n X → , b – constant ⇒ b P n X → 1. Let X 1 , X 2 , … , X n be a random sample of size n from the distribution with probability density function f ( x ; θ ) = ≤ ≤- ⋅ otherwise 1 1 θ θ 1 θ x x 0 < θ < ∞ ....
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10_19 - STAT 410 Examples for Fall 2011 Let X n be a...

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