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Unformatted text preview: s for this upper bound
to be less than δ . According to this suﬃcient condition, we set n large enough so that
1
≤δ
4nε 2
Plugging in δ = 0.01 and ε = 0.05, we obtain nmin = 10000.
M. Chen (IE@CUHK) ENGG2430C lecture 10 22 / 29 Beyond Convergence in Probability 1
The Law of Large Number says that the sample mean Yn = n ∑n=1 Xi
i
is very likely to be close to the true mean E [X ] as n goes large. Xi can follow any distribution, with ﬁnite mean. The Central Limit Theorem further says that Yn approximately
follows Gaussian distribution as n goes large. (Stronger)
Xi can follow any distribution, with ﬁnite mean and variance. M. Chen (IE@CUHK) ENGG2430C lecture 10 23 / 29 Central Limit Theorem (CLT) Let X1 , X2 , . . . be an inﬁnite sequence of independent identically
distributed r.v.s with mean µ and variance σ 2 . Deﬁne a r.v.
Zn = X1 + X2 + · · · + Xn − n µ
√
.
σn Then the CDF of Zn converges to the standard normal CDF
N (0, 1):
ˆz
2
1
Φ(z ) = √
e −x /2 dx ,
2π −∞
in the sense that
lim P (Zn ≤ z ) = Φ(z ), n→∞ for every z . Xi can follow any distributi...
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This document was uploaded on 03/31/2014.
 Spring '14

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