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Unformatted text preview: Central Limit Theorem Notation . N ( μ,σ 2 ) refers to normal with mean μ and variance σ 2 . Have just seen that when X 1 ,...,X n } is a ran dom sample from N ( μ,σ 2 ) then sample means ¯ X ∼ N ( μ,σ 2 /n ) . This follows from a more gen eral result that: If X i ∼ N ( μ i , σ 2 i ) are independent normals then any linear combination Y = a + n X i =1 a i X i is also normal. By rules of linear combinations E[ Y ] = a + n X i =1 a i μ i Var[ Y ] = n X i =1 a 2 i σ 2 i 1 The previous example used the spe cial case where all μ i are same μ, all standard deviations σ i are the same σ , a = 0 , all other a i are the same 1 n . Then Y becomes the sample mean ¯ X = 1 n n X i =1 X i with E[ ¯ X ] = a + n X i =1 a i μ i = 0 + n X i =1 1 n μ = μ and Var[ ¯ X ] = n X i =1 a 2 i σ 2 i = n X i =1 ( 1 n ) 2 σ 2 = σ 2 n 2 That is ¯ X ∼ N ( μ,σ 2 /n ) By standardizing, the random variable Z = ¯ X μ σ/ √ n is standard normal. Another special case when all μ i = μ and all σ i = σ is the sample sum T = n X i =1 X i so T ∼ N ( nμ,nσ 2 ) which standardizes using Z = T nμ σ √ n 3 But what if { X 1 ,...,X n } is a random sample from a nonnormal population? Then no general statement (except for a few special cases) can be made about the exact distribution of a linear com bination, including a sample mean ¯ X or sample sum T except when n is large. 4 Central Limit Theorem. When n is “large” there is a famous result known as the Central Limit Theorem, which states that if { X 1 ,...,X n } is a random sample from any population with mean μ and standard deviation σ then ¯ X ≈ N ( μ,σ 2 /n ) T ≡ n X i =1 X i ≈ N ( nμ,nσ 2 ) In other words sums and averages are approximately normal for large n....
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This note was uploaded on 01/03/2012 for the course EE 1244 taught by Professor Drera during the Fall '10 term at Conestoga.
 Fall '10
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