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# sta 2006 0809_addition_example_1 - the Poisson distribution...

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STA2006 Basic Concepts in Statistics II (2008-09, 2nd term) Additional Example 1 1. Let X 1 , X 2 , ... be iid random variables with E ( X i ) = μ and V ar ( X i ) = σ 2 < . Define ¯ X n = (1 /n ) n i =1 X i . Prove, for every > 0, lim n →∞ P ( | ¯ X n - μ | ≥ ) = 0 , that is, ¯ X n converges in probability to μ . 2. Let the random variable Y n have a distribution that is Bin ( n, p ). Prove that (a) Y n /n converges to p in probability, (b) 1 - Y n /n converges to 1 - p in probability.
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Unformatted text preview: the Poisson distribution. Let X have the hypergeometric distribution P ( X = x | N,M,K ) = ± M x ² ± N-M K-x ² ± N K ² , x = 0 , 1 ,...K. Show that as N → ∞ ,M → ∞ and M/N → p , P ( X = x | N,M,K ) → ³ K x ! p x (1-p ) K-x , x = 0 , 1 ,...,K....
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