E yn varyn 2 n apply chebyshev inequality

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Unformatted text preview: infinite) The probabilities of a series of events, {Yn lies in [µ − ε , µ + ε ]}, converge to 1 M. Chen ([email protected]) ENGG2430C lecture 10 15 / 29 Good proof motivates and consolidates intuition Proof. [The finite variance case] Let X1 , X2 , . . . be a infinity sequence of i.i.d. random variables with mean µ and variance σ 2 . E [Yn ] = µ , Var(Yn ) = σ 2 /n. Apply Chebyshev inequality to Yn , we have for any ε > 0, P (|Yn − µ | ≥ ε ) ≤ σ2 Var(Yn ) = 2; ε2 nε therefore, lim P (|Yn − µ | ≥ ε ) = 0. n→∞ M. Chen ([email protected]) ENGG2430C lecture 10 16 / 29 Insights from the Law of Large Number 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. We also have a Strong Law of Large Number. (Advanced topic.) Allow us to estimate the mean of a distribution. The aggregate of a large number of independent randomness (whose complete des...
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