Hw9Sol_9_b - SIEO 3600 (IEOR Majors) Solution to Assignment...

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Unformatted text preview: SIEO 3600 (IEOR Majors) Solution to Assignment #9 Problem 9(b) Introduction to Probability and Statistics April 11, 2010 Solution to Assignment #9 Problem 9(b) 9(b) ( Sum of independent normals is again normal ) Suppose X 1 Normal ( 1 , 2 1 ), X 2 Normal ( 2 , 2 2 ). Assume X 1 and X 2 are independent. Show that Z X 1 + X 2 Normal ( 1 + 2 , 2 1 + 2 2 ), i.e., Z is normally distributed with mean 1 + 2 and variance 2 1 + 2 2 . Hint: Evaluate the CDF of Z , that is F Z ( z ) = P ( Z z ) = P ( X 1 + X 2 x ). Then differen- tiate the CDF to obtain its PDF and show that this is indeed the PDF of a Normal ( 1 + 2 , 2 1 + 2 2 ). Note that P ( X 1 + X 2 x ) can be computed by a double integral over a certain region in R 2 . Solution: To simplify the computation, note that X 1 + X 2 = 1 X 1- 1 1 + X 2- 2 1 + ( 1 + 2 ). Recall that linear transformation of normal is again normal, i.e., Y = aX + b Normal...
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Hw9Sol_9_b - SIEO 3600 (IEOR Majors) Solution to Assignment...

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