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Unformatted text preview: , Var(Xi ) = p (1 − p ).
The sample mean is
Yn = 1n
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.
n The sequence of 1 , n = 1, 2, . . . converges to 0.
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
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- Spring '14