This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 1. Problem 7.2.8 in the text. Solution (a) The bias of the estimate is bias = E ( p ) p = E ( X/ 11) p = 10 11 p p = p/ 11 (b) The variance of the estimate is Var( p ) = Var( X/ 11) = 1 121 Var( X ) = 10 p (1 p ) 121 (c) The mean square error is MSE = E ( p p ) 2 = Var( p ) + bias 2 = 10 p (1 p ) 121 + p 2 121 = 10 p 9 p 2 121 (d) The bias of the estimator p 1 = X/ 10 is E ( p 1 ) p = E ( X/ 10) p = 0 The mean square error is therefore MSE( p 1 ) = Var( p 1 ) + bias 2 = 1 100 Var( X ) = p (1 p ) 10 The di erence between mean square errors is given by 10 p 9 p 2 121  {z } MSE for p p (1 p ) 10  {z } MSE for p 1 = 31 p 2 21 p 1210 we want to know when the quantity 31 p 2 21 p 1210 is negative, which only happens when 31 p 2 21 p is negative; in other words, 31 p 2 21 p p (31 p 21) 31 p 21 p 21 / 31 2. Suppose that a 6sided die is rolled n times, showing values X 1 ,...,X n . Make a point estimate of the probability of rolling a 1 on the die, and write the standard error of this estimate. 1 Solution The number of 1 's shown on the die after n rolls is a B ( n,p = 1 / 6) random variable, which we can denote by Y . From lecture 11, slides 79, our estimate is the sample proportion p = Y/n , which has the approximate distribution p N p, p (1 p ) n = N 1 / 6 , 5 36 n 3. Determine the maximum likelihood estimator of when X 1 ,...,X n is a random variable with density function f ( x ) = 1 2 e x  for < x < . Solution Write the likelihood function L ( x 1 ,...,x n ; ) = 1 2 e x 1  1 2 e x n  = 2 n e (  x 1  + +  x n  ) The loglikelihood function is ( x 1 ,...,x n ; ) = log 2 n e (  x 1  + +  x n  ) = n log 2  x 1     x n  We want to maximize this function, which is a bit sticky because of the absolute values. Note that d d  x i  = ( 1 if < x i 1 if > x i and therefore d d = ( #entries where > x i ) ( #entries where < x i ) To maximize this function we set...
View Full
Document
 Spring '08
 Staff

Click to edit the document details