# week12 - Law of Large Numbers Toss a coin n times Suppose 1...

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week 12 1 Law of Large Numbers Toss a coin n times. Suppose X i ’s are Bernoulli random variables with p = ½ and E ( X i ) = ½. The proportion of heads is . Intuitively approaches ½ as n Æ . = T up came toss i if H up came toss i if X th th i 0 1 = = n i i n X n X 1 1 n X

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week 12 2 Markov’s Inequality If X is a non-negative random variable with E ( X ) < and a >0 then, () ( ) a X E a X P
week 12 3 Chebyshev’s Inequality For a random variable X with E ( X ) < and V ( X ) < , for any a >0 Proof: () ( ) 2 a X V a X E X P

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week 12 4 Back to the Law of Large Numbers Interested in sequence of random variables X 1 , X 2 , X 3 , … such that the random variables are independent and identically distributed (i.i.d). Let Suppose E ( X i ) = μ , V ( X i ) = σ 2 , then and Intuitively, as n Æ , so = = n i i n X n X 1 1 () μ = = = = = n i i n i i n X E n X n E X E 1 1 1 1 n X V n X n V X V n i i n i i n 2 1 2 1 1 1 σ = = = = = ( ) 0 n X V ( ) = n n X E X
week 12 5 Formally, the Weak Law of Large Numbers (WLLN) states the following: Suppose X 1 , X 2 , X 3 , …are i.i.d with E ( X i ) = μ < , V ( X i ) = σ 2 < , then for any positive number a as n Æ .

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week12 - Law of Large Numbers Toss a coin n times Suppose 1...

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