stat400lec20 - normal distribution N( , 2 ), then...

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Statistics 400 Section 6.2 and 6.3 Distribution of sums of the independent random variables Theorem 6.2.3 If X1, X2, …, Xn are n independent random variables with respective means μ 1, μ 2, …, μ n and variances 2 1 σ , 2 2 , …, 2 n , then the mean and variance of Y= n i Xi a 1 , i n i Y a μ = 1 , = n i i Y a 1 2 2 2 Theorem 6.2-4 If X1, X2, …, Xn are n independent random variables with respective moment-generating function M Xi (t), then the moment-generating function of Y= n i Xi a 1 is ) ( ) ( 1 t a M t M i X n Y i Π = , Two special cases Ping Ma Lecture 20 Fall 2005 - 1 -
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Important fact, Typically, distribution has a unique moment-generating function. Moment-generating function distribution If X1, X2,X3 are observations of a random sample of size 3 from an exponential distribution with mean θ , then distribution of the sample mean X is If X1, X2, …, Xn are observations of a random sample of size n from a
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Unformatted text preview: normal distribution N( , 2 ), then distribution of the sample mean X is Ping Ma Lecture 20 Fall 2005- 2 -Theorem 6.3-2 Let the distribution of independent random variables X1, X2, , Xn be 2 (r1), 2 (r2),, 2 (rn), respectively, then distribution of Y= X1+X2+ +Xn is 2 (r1+r2++rn) Theorem 6.3-3 If Z1, Z2, , Zn are observations of a random sample of size n from a standard normal distribution N(0,1), , then W= Z1 2 +Z2 2 + +Zn 2 ~ 2 (n) Ping Ma Lecture 20 Fall 2005- 3 -If X1, X2, , Xn are observations of a random sample of size n from a normal distribution N( , 2 ), then we have (a) X and S 2 are independent (b) 2 2 ) 1 ( S n-is 2 (n-1) Ping Ma Lecture 20 Fall 2005- 4 -...
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stat400lec20 - normal distribution N( , 2 ), then...

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