notes-09-03

notes-09-03 - the observations we made last time: the...

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M4056 Lecture Notes. September 3, 2010 Sampling from a normal distribution Recall: If X is n (0 , 1), M X ( t ) = e t 2 / 2 M aX + b ( t ) = e bt M X ( at ) If Y is n ( μ, σ 2 ), then Y = σX + μ , where X is n (0 , 1). Thus, if Y n ( μ, σ 2 ) , M Y ( t ) = e μ t + σ 2 t 2 / 2 . Suppose X 1 , . . ., X n are any random variables. Then M X ( t ) = E e t X = E e ( t/n )( X 1 + ··· + X n ) = M X 1 + ··· + X n ( t/n ) . We have seen previously that the mgf of distribution of a sum of independent random variables is the product of the mfg s of the summands. Hence: if X 1 , . . ., X n are iid X , then M X ( t ) = [ M X ( t/n )] n . (This appears in your book as Theorem 5.2.7 .) This enables us to calculate the distribution of the sum of iid normal variables. If X 1 , . . ., X n are iid n ( μ, σ 2 ), then M X ( t ) = [ e μ ( t/n )+ σ 2 ( t/n ) 2 / 2 ] n = e n [ μ t/n + σ 2 ( t/n ) 2 / 2] = e μ t +( σ 2 /n ) ( t 2 / 2) The last expression is the mgf of an n ( μ, σ 2 /n ) random variable. This is consistent with
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Unformatted text preview: the observations we made last time: the expected value and variance of X are as we have already seen they must be. But even more: the sample mean is itself normal, so these two numbers tell the whole story. We restate this important result: Fact. If X 1 , . . ., X n are iid n ( , 2 ), then X is n ( , 2 /n ). (5.3.1.b) Next, we turn to the distribution of the sample variance S 2 when samples of size n are drawn from an n ( , 2 ) distribution. Fact. X and S 2 are independent. (5.3.1.a) Fact. ( n-1) S 2 / 2 has a 2 n-1 (chi squared with n-1 degrees of freedom) distribution. (5.3.1.c)...
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This note was uploaded on 11/29/2011 for the course MATH 4056 taught by Professor Staff during the Fall '08 term at LSU.

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