stat1801_6

# stat1801_6 - PopulationandSample Population 2 X,S2...

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Population and Sample Population Sample 2 , σ μ Population parameters 2 , S X Sample Statistics 2 , S X 2 , Inference Sampling distribution?

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Population and Sample Population 6 = μ 8 2 = σ Sample (with replacement) Sample Mean Sample Variance { } 2 1 , X X 2 2 1 X X X + = ( 29 ( 29 { } 2 2 2 1 2 1 2 1 X X X X S - + - - = ( 29 2 2 2 1 2 X X S - = ………………. 2 0 Prob = 1/25 3 2 Prob = 2/25 6 8 Prob = 2/25
Population and Sample 0.08 9 10 8 7 6 5 4 3 2 0.04 0.12 0.16 0.2 0.16 0.12 0.08 0.04 Prob X Sampling distributions 0.08 0.16 0.24 0.32 0.2 Prob 32 18 8 2 0 2 S ( 29 04 . 0 10 08 . 0 3 04 . 0 2 × + + × + × = X E μ = 6 ( 29 08 . 0 32 32 . 0 2 2 . 0 0 2 × + + × + × = S E 2 8 σ = Unbiased ( 29 ( 29 2 4 = = X Var X Var ( 29 ( 29 295 . 9 4 . 86 2 2 = = S Var S Var

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Random Sample  { } n X X X ,..., , 2 1 random sample Each X drawn from same population ( f ( x ) ) X’s are independent iid = = n i i X n X 1 1 Sample Mean ( 29 = - - = n i i X X n S 1 2 2 1 1 Sample Variance ( 29 μ = X E ( 29 n X Var 2 σ = ( 29 2 2 = S E
Random Sample  { } n X X X ,..., , 2 1 random sample ( 29 μ = X E ( 29 n X Var 2 σ = Example : Final examination scores 4 . 73 = 16 . 237 2 = For a RS of 4 students : ( 29 4 . 73 = = μ X E 7 . 7 4 16 . 237 = = = n SE For a RS of 16 students : ( 29 4 . 73 = = μ X E 85 . 3 16 16 . 237 = = = n SE ( 29 ( 29 n X Var X se = = standard error

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Sampling Distribution  { } n X X X ,..., , 2 1 random sample Moment generating function of sample mean ( 29 n X X n t M t M = Example ( 29 λ Exp X X X iid n ~ ,..., , 2 1 ( 29 < - = t t t M X , ( 29 n t t n n n t t M n n X < - = - = , ( 29 n n X , ~ Γ
Chi-square Distribution  ( 29 ( 29 0 , 2 2 1 2 1 2 2 Γ = - - x e x r x f x r r ( 29 r X E = ( 29 r X Var 2 = 2 2 1 , 2 r r χ Γ

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Chi-square Distribution  2 16 ~ χ X ( 29 99 . 0 32 = X P 2 13 ~ X ( 29 05 . 0 892 . 5 = X P 2 8 X ( 29 95 . 0 ? = X P 51 . 15
Chi-square Distribution  Example: Claim sizes (\$m) ( 29 4 . 0 , 2 ~ ,..., , 2 1 Γ iid n X X X Total payment of 15 claims = = 15 1 i i X S ( 29 4 . 0 , 30 ~ Γ S Claim reserve to have 95% chance to cover total payment = ? Find c such that P ( S c ) = 0.95 ( 29 2 60 5 . 0 , 30 ~ 8 . 0 χ Γ S ( 29 95 . 0 8 . 0 8 . 0 = c S P 08 . 79 8 . 0 2 60 , 05 . 0 = = χ c 85 . 98 = c

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Chi-square Distribution  ( 29 2 1 2 2 2 1 ~ , ~ ,..., , n n i i iid n X N X X X χ σ μ = - ( 29 2 1 2 2 1 , 2 1 ~ 1 , 0 ~ Γ Z N Z Chi-square distribution is closely related to normal distribution.
Sampling Distribution Population { } 2 1 , X X 0.08 9 10 8 7 6 5 4 3 2 0.04 0.12 0.16 0.2 0.16 0.12 0.08 0.04 Prob X Sampling distributions 0.08 0.16 0.24 0.32 0.2 Prob 32 18 8 2 0 2 S Random Sample

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Sampling Distribution Population { } 2 1 , X X RS Dist. of 2 S 2 S X Dist. of X
Normal Population Population ( 29 2 , σ μ N { } n X X X ,..., , 2 1 RS Dist. of X n N 2 , X Dist. of ( 29 2 2 1 S n - 2 1 - n χ 2 S independent

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Normal Population ( 29 2 2 1 , ~ ,..., , σ μ N X X X iid n
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## This note was uploaded on 02/01/2012 for the course STAT 1801 taught by Professor Mrchung during the Fall '10 term at HKU.

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stat1801_6 - PopulationandSample Population 2 X,S2...

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