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Unformatted text preview: Econ 714  Midterm Solutions Lorenzo Braccini * October 28, 2011 Question 1 Let X be a Bernoulli distributed Random Variable with parameter p . Define Z = 3 X 1. First note that Z is a Random Variable taking values on the set { , 2 } with probability: P( Z = 0) = P(3 X 1 = 0) = P(3 X = 1) = P( X = 0) = 1 p and P( Z = 2) = P(3 X 1 = 2) = P(3 X = 3) = P( X = 1) = p a) By definition of the expectation operator we can write: E [ Z ] = X z ∈{ , 2 } z P( Z = z ) = 0(1 p ) + 2 p = 2 p b) By a similar reasoning we have: E Z 2 = X z ∈{ , 2 } z 2 P( Z = z ) = 0(1 p ) + 4 p = 4 p * [email protected] 1 c) The variance of Z is defined as: Var( Z ) = E ( Z E [ Z ]) 2 = E Z 2 E [ Z ] 2 = 4 p 4 p 2 = 4[ p (1 p )] = 4Var( X ) d) First note that: ˆ p = 1 n n X i =1 x i = ¯ X n Then by the definition of bias we can write: Bias (ˆ p ) = E [ˆ p ] p = E ¯ X n p = 1 n n X i =1 E [ X i ] p = 1 n n X i =1 p p = p p = 0 Hence the estimator ˆ p is unbiased....
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This note was uploaded on 02/29/2012 for the course GG 101 taught by Professor Gg during the Spring '12 term at UPenn.
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