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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 + 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 ) 2 V ar n X i =1 X i + n 2 ! = np (1 - p ) ( n + n ) 2 MSE ( ˆ p 2 ) = np (1 - p ) ( n + n ) 2 + n 2 - np n + n ! 2 = 1 4( n + 1) 2 1
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0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.01 0.02 0.03 0.04 0.05 p MSE MSE1 n=5 MSE1 n=10 MSE1 n=100 MSE2 n=5 MSE2 n=10 MSE2 n=100 2. Suppose we roll a die 1000 times independently, and we observe the number six 180 times. Let p be the probability that we observe the number six each time.
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