MIT2_854F10_useful

MIT2_854F10_useful - Useful Distributions Chuan Shi Suppose...

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± ± Useful Distributions Chuan Shi Suppose we have a population X N ( µ, σ 2 ), and x 1 , x 2 , , x n is a n size sample of the · · · population. The mean and variance of the sample are given as: n n 1 ² 1 ² x ¯ = x i and s 2 = ( x i x ¯) 2 (1) n i =1 n 1 i =1 The following distributions constructed with ¯ x and s 2 (or s ) are very useful in Statistics. They could be helpful for your homework. ¯ N µ, σ 2 or x ¯ µ N (0 , 1) (2) x σ n n x ¯ s µ t ( n 1) (3) n ( n σ 2 1) s 2 χ 2 ( n 1) (4) Suppose there are two populations X N ( µ 1 , σ 2 ) and Y N ( µ 2 , σ 2 ), and a n 1 size sample of 1 2 X , and a n 2 size sample of Y . Means of the two samples are ¯ x and y ¯, respectively. Variances of the two samples are s 2 1 and s 2 2 , respectively. Then,
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MIT2_854F10_useful - Useful Distributions Chuan Shi Suppose...

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