The weak law of large number makes this argument

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Unformatted text preview: , Var(Xi ) = p (1 − p ). The sample mean is Yn = 1n Xi n i∑ =1 1 Its mean and variance are: E [Yn ] = p , Var(Yn ) = n p (1 − p ) Thus, as n increases, Yn is likely to take values around p . The Weak Law of Large Number makes this argument precise and generalizes it. M. Chen (IE@CUHK) ENGG2430C lecture 10 12 / 29 Convergence of a sequence of numbers Let a1 , a2 , . . . be a sequence of real numbers, and let a be another real number. We say that the sequence an converges to a, or limn→∞ an = a, if for every ε > 0, there exists some n0 such that |an − a| ≤ ε , ∀n ≥ n0 . Intuition: if limn→∞ an = a, then for any given accuracy level ε , an must be within ε of a, when n is large enough. Examples: n The sequence of 1 , n = 1, 2, . . . converges to 0. 2 The sequence of probabilities converges to 0. M. Chen (IE@CUHK) ENGG2430C lecture 10 13 / 29 Convergence of a sequence of random variables Let Y1 , Y2 , . . . be a sequence of random variables, and let a be another...
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