stat400lec20

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

This preview shows pages 1–4. Sign up to view the full content.

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 -

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 -...
View Full Document

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online