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Lecture 13 - Sum of i.i.d random variables

# Lecture 13 - Sum of i.i.d random variables - 2 MOMENT...

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Lecture 13 ORIE3500/5500 Summer2011 Li 1 Sum of i.i.d. random variables Let X be independent and identically distributed (i.i.d.) random variables with expected value μ and variance σ 2 . Let ¯ X = n - 1 n i =1 X i . We have E ¯ X = μ , var ( ¯ X ) = σ 2 /n and cov ( X i - ¯ X, ¯ X ) = 0. Proof: E ¯ X = n - 1 n X i =1 EX i = μ. var ( ¯ X ) = n - 2 n X i =1 var ( X i ) = σ 2 /n. cov ( X i - ¯ X, ¯ X ) = cov ( X i ,n - 1 n X j =1 X j ) - var ( ¯ X ) = n - 1 n X j =1 cov ( X i ,X j ) - σ 2 /n = 0 . A related question: What if X i ’s are not independent? The idea of diversiﬁcation: var ( X + Y ) = var ( X )+ var ( Y )+2 cov ( X,Y ). If X and Y negatively correlated, then var ( X + Y ) < var ( X ) + var ( Y ). 2 Moment Generating Function For a random variable X , the moment generating function of X is deﬁned as φ X ( t ) = E ( e tX ) . There is a reason for this name. If one is given the moment generating function of a random variable then one can get all the moments of the random variable. If we evaluate the n th derivative of the moment generating function at 0, we will get the n th moment of X , that is, d n φ X ( t ) dt n t =0 = E ( X n ) . This is because d n φ X ( t ) dt n = d n E ( e tX ) dt n = E ( d n e tX ) dt n = E ( X n e tX ) . 1

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Lecture 13 - Sum of i.i.d random variables - 2 MOMENT...

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