STAT100B_HW1S

STAT100B_HW1S - STAT 100B Homework 1 Solutions Denise Tsai...

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STAT 100B: Homework 1 Solutions Denise Tsai January 31, 2011 1. Let X 1 ,...,X n Bernoulli( p ) independently. Let S = n i =1 X i . Suppose we have two estimators of p . (1) ˆ p = S/n , and (2) ˆ p = ( S + n/ 2) / ( n + n ). (a) Calculate the bias and variance of each estimator. (b) Calculate the mean squared error of each estimator. Plot the mean squared errors of the two estimators together over the true value of p [0 , 1], for n = 5, n = 10, and n = 100 respectively. Solution: Denote ˆ p 1 = S n : E ( ˆ p 1 ) = E ( S n ) = 1 n E n X i =1 X i ! = 1 n ( np ) = p Bias ( ˆ p 1 ) = E p ) - p = p - p = 0 V ar ( ˆ p 1 ) = V ar n X i =1 X i ! = 1 n 2 × n × p (1 - p ) = p (1 - p ) n MSE ( ˆ p 1 ) = V ar p ) + Bias 2 p ) = p (1 - p ) n Denote ˆ p 2 = S + n 2 n + n E ( ˆ p 2 ) = E ±∑ n i =1 X i n + n ² + n 2 n + n = np + n 2 n + n Bias ( ˆ p 2 ) = np + n 2 n + n - p = n 2 - np n + n V ar ( ˆ p 2 ) = 1 ( n + n
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This note was uploaded on 09/20/2011 for the course STAT 100B taught by Professor Wu during the Winter '11 term at UCLA.

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STAT100B_HW1S - STAT 100B Homework 1 Solutions Denise Tsai...

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