a1s - ECON 3210 3.0 Section A Fall 2011 Solution for...

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ECON 3210 3.0 Section A - Fall 2011 Solution for Assignment 1 Question 1: a. Assume X is a discrete random variable with probability function p ( x ) = P ( X = x ). Then E ( X ) = μ x = xp ( x ) and var ( X ) = σ 2 x = E (( X - μ x ) 2 ). Since Y = a + bX , then E ( Y ) = μ y = E ( a + bX ) = X ( a + bx ) p ( x ) = a X p ( x ) + b X xp ( x ) = a + x . Similarly, if X is a continuous random variable with density function f ( x ), then E ( X ) = μ x = R xf ( x ) dx and var ( X ) = σ 2 x = E (( X - μ x ) 2 ). Since Y = a + bX , then E ( Y ) = μ y = E ( a + bX ) = Z ( a + bx ) f ( x ) dx = a Z f ( x ) dx + b Z xf ( x ) dx = a + x . b. Regardless if X is discrete or continuous, we have Y - μ y = ( a + bX ) - ( a + x ) = b ( X - μ x ). Therefore ( Y - μ y ) 2 = b 2 ( X - μ x ) 2 and hence var ( Y ) = E (( Y - μ y ) 2 ) = E ( b 2 ( X - μ x ) 2 ) = b 2 E (( X - μ x ) 2 ) = b 2 var ( X ) . Thus
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This note was uploaded on 02/05/2012 for the course ECON 4140 taught by Professor A during the Spring '11 term at York University.

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a1s - ECON 3210 3.0 Section A Fall 2011 Solution for...

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