W10 - Lecture 28 Outline and Examples Weak Law of Large...

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Unformatted text preview: Lecture 28- Outline and Examples Weak Law of Large Numbers (Ross Â§ 8.2) Central Limit Theorem.(Ross Â§ 8.3) Theorem. (Weak Law of Large Numbers) Let X 1 , X 2 , .. . , be a sequence of independent and identically distributed random variables, each having finite (common) mean E [ X i ] = Î¼ . Then, for any > 0, P X 1 + X 2 + Â·Â·Â· + X n n- Î¼ â‰¥ â†’ as n â†’ âˆž . Theorem. (Weak Law of Large Numbers) Let X 1 , X 2 , .. . , be a sequence of independent and identically distributed random variables, each having finite (common) mean E [ X i ] = Î¼ . Then, for any > 0, P X 1 + X 2 + Â·Â·Â· + X n n- Î¼ â‰¥ â†’ as n â†’ âˆž . Proof. Theorem. (Weak Law of Large Numbers) Let X 1 , X 2 , .. . , be a sequence of independent and identically distributed random variables, each having finite (common) mean E [ X i ] = Î¼ . Then, for any > 0, P X 1 + X 2 + Â·Â·Â· + X n n- Î¼ â‰¥ â†’ as n â†’ âˆž . Proof. Recall example done earlier where we calculated the expected value and variance of the â€™ sample mean â€™: E X 1 + X 2 + Â·Â·Â· +...
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W10 - Lecture 28 Outline and Examples Weak Law of Large...

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