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Unformatted text preview: on, with ﬁnite mean AND variance.
M. Chen ([email protected]) ENGG2430C lecture 10 24 / 29 Example
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Let Xi be i.i.d. Bernoulli rvs. We plot the PMF of Y = n ∑n=1 Xi .
i M. Chen ([email protected]) ENGG2430C lecture 10 25 / 29 CLT is important in engineering and ﬁnance 1
The Central Limit Theorem says that the sample mean Yn = n ∑n=1 Xi
i
approximately follows Gaussian distribution as n goes large. (Strong) Xi can follow any distribution, with ﬁnite mean and variance.
Allow one to characterize the distribution of the aggregate of a large
number of independent randomness, even though we don’t know the
individual distributions! Because these reasons, CLT is very useful in modeling the aggregate
random eﬀects in many engineering and ﬁnance scenarios.
Noise in wireless communications.
Video streaming rate (high rate leads to better video quality) in a P2P
video streaming system.
Daily index increment/decrement in a stock market. M. Chen ([email protected]) ENGG2430C lecture 10 26 / 29 ATV poll problem: The CLT approach
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Let r.v. Yn = n ∑n=1 Xi be the estimate of p
i
−
Yn roughly follows N (p , p (1n p ) ). Search for δ so that P (Yn ∈ [p − δ , p + δ ]) ≥ 0.99,
δ is a function of n. Let δ = ε solve for n ( messy!)
USEFUL FACT: For a Gaussian variable Z with mean µ and variance σ 2 ,
P (Z ∈ [µ − 3σ , µ + 3σ ]) ≈ 0.99. (From wikipedia)
M. Chen ([email protected]..
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This document was uploaded on 03/31/2014.
 Spring '14

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