Midterm 1 + 2

Midterm 1 + 2 - Econ 714 Midterm Solutions Lorenzo Braccini...

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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 { 0 , 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 ∈{ 0 , 2 } z P( Z = z ) = 0(1 - p ) + 2 p = 2 p b) By a similar reasoning we have: E Z 2 = X z ∈{ 0 , 2 } z 2 P( Z = z ) = 0(1 - p ) + 4 p = 4 p * [email protected] 1
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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 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. e) Again by its definition, the bias of 4ˆ p as an estimator for E [ Z 2 ] is given by: Bias (4ˆ p ) = E [4ˆ p ] - E Z 2 = 4 E ¯ X n - 4 p = 4 p - 4 p = 0 Hence the estimator 4ˆ p for E [ Z 2 ] is unbiased.
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