5102-Lecture-09

5102-Lecture-09 - Lecture 9 Stat 5102-004 7 February 2011...

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Unformatted text preview: Lecture 9 Stat 5102-004 7 February 2011 Recall from last time . . . 1. If Y 1 , . . . , Y n iid ∼ N( μ, σ 2 ) and both μ and σ 2 are unknown, then (a) the mle of μ is ˆ μ = ¯ Y , and (b) the mle of σ 2 is ˆ σ 2 = 1 n ∑ n i =1 ( Y i- ¯ Y ) 2 . 2. Estimators are statistics: they have sampling distributions. Y 1 , . . . , Y n iid ∼ N( μ, 1) √ n ( ¯ Y- μ ) ∼ N(0 , 1) Y 1 , . . . , Y n iid ∼ Gamma(1 , β ) n ¯ Y β ∼ Gamma( n , 1) How is (ˆ μ, ˆ σ 2 ) distributed? STAT 5102 (Theory of Statistics) Lecture 9 1 / 22 Square of a Standard Normal Random Variable Let Z ∼ N (0 , 1) and W = Z 2 . F W ( w ) = Pr( W ≤ w ) = Pr( | Z | ≤ √ w ) = 2Pr(0 ≤ Z ≤ √ w ) = 2 F Z ( √ w )- 1 2 = 2 F Z ( √ w )- 1 z Pr ( Z ≤ z 29-3-1 1 3 1 w Pr(|Z| ≤ w) 1 2 3 1 w Pr ( W ≤ w 29 3 6 9 1 STAT 5102 (Theory of Statistics) Lecture 9 2 / 22 Square of a Standard Normal Random Variable Differentiating F W ( w ) = 2 F Z ( √ w )- 1 gives the density f W ( w ) = f Z ( √ w ) w- 1 2 = 1 √ 2 π e- 1 2 w w- 1 2 = (1 / 2) 1 / 2 Γ(1 / 2) w 1 2- 1 e- 1 2 w over the positive reals. The random variable W has a Gamma( 1 2 , 1 2 ) distribution. Gamma( 1 2 , 1 2 ) ∼ χ 2 (1) STAT 5102 (Theory of Statistics) Lecture 9 3 / 22 Properties of the χ 2 Family The χ 2 family is a subfamily of gammas closed under addition. W 1 , . . . , W n iid ∼ χ 2 (1) = ⇒ W = ∑ n i =1 W i ∼ χ 2 ( n ) W ∼ Gamma( n 2 , 1 2 ) E W = n Var W = 2 n A χ 2 with 2 degrees of freedom is an exponential distribution....
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This note was uploaded on 02/24/2011 for the course STAT 5102 taught by Professor Staff during the Spring '03 term at Minnesota.

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5102-Lecture-09 - Lecture 9 Stat 5102-004 7 February 2011...

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