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Unformatted text preview: NOTES FOR STAT 5102004 LECTURE 13 16 FEBRUARY 2011 1. Predictive Region for a Normal with known variance Let Y 1 ,...,Y n +1 iid N( , 2 ) with 2 known. Suppose we observe y = ( y 1 ,...,y n ), and want to make predictions concerning Y n +1 . Note that Y n +1 Y n N(0 , 2 (1 + 1 n )) since Y n and Y n +1 are independent normals with common mean . Thus, from r n n + 1 1 ( Y n +1 Y n ) N(0 , 1) we can use equalities such as Pr Y n q 1 + 1 n 1 . 96 Y n +1 Y n + q 1 + 1 n 1 . 96 = 0 . 95 to create level prediction intervals. Specifically, y n q 1 + 1 n 1 . 96 is a level 0 . 95 prediction interval for y n +1 . By comparison, recall that Pr Y n q 1 n 1 . 96 Y n + q 1 n 1 . 96 = 0 . 95 led to a 95% confidence interval for . The difference between a confidence interval and a prediction interval is the difference between trying to cover a fixed quantity and trying to cover a random quantity. Prediction always has to account for the sampling variability in a new observation. 2. Posterior for with known 2.1. Normal Conjugate Prior. Let Y 1 ,...,Y n iid N( , 1 ) with known. Assume a normal prior for with mean and precision . The prior density is ( ) = (2 ) 1 2 1 2 1 2 exp ( 1 2 (  ) 2 ) . The likelihood function satisfies lik(  y ) n 2 exp ( 1 2 n i =1 ( y i ) 2 ) which, letting n i =1 ( y i y ) 2 = k (I P) y k 2 , can be separated out into lik(  y ) exp ( 1 2 n (  x ) 2 ) n 2 exp ( 1 2 k (I P) y k 2 ) . (1) 1 2.2. Approaching the Posterior. The posterior density is proportional to the prior den...
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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.
 Spring '03
 Staff
 Statistics, Variance

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