week04-2

week04-2 - TA session 4 Econ. 103, winter 2010 Wed., Jan....

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TA session 4 Econ. 103, winter 2010 Wed., Jan. 27, 2010, 10:00 a.m. and 1:00 p.m. in PP2400E. 2 The distribution of X i is not the distribution of X This is something people have trouble with at first. Let X 1 ∼ N ( μ,σ 2 ) Suppose μ = 0 and σ = 1. Then the distribution of X 1 looks something like Now, in addition to X , let X 2 ∼ N ( μ,σ 2 ) Note that X 2 has the same mean and variance as X 1 ! From the rules I gave you in my week 3 lecture, you know that Var ± X 1 + X 2 2 ² = σ 2 2 2 + σ 2 2 2 = σ 2 2 Hence ( X 1 + X 2 ) / 2 ∼ N ( μ,σ 2 / 2): 1
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2.1 Moral of the story The variance of several random variables with the same distribution added together is X 1 + X 2 + X 3 + ... + X n n ∼ N ( μ,σ 2 /n ) and if one were to graph this distribution, it would tend to collapse around the mean as n → ∞ . Here are several distributions with μ = 0 using n = 1 , 2 , 4 , 8: Now the estimate of the mean is just X i = N n =1 X i n and having endured the above, you should recognize its distribution
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week04-2 - TA session 4 Econ. 103, winter 2010 Wed., Jan....

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