Hw9Sol_9_b

# Hw9Sol_9_b - SIEO 3600(IEOR Majors Solution to Assignment#9...

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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 ( μ 1 , σ 2 1 ), X 2 ∼ Normal ( μ 2 , σ 2 2 ). Assume X 1 and X 2 are independent. Show that Z ≡ X 1 + X 2 ∼ Normal ( μ 1 + μ 2 , σ 2 1 + σ 2 2 ), i.e., Z is normally distributed with mean μ 1 + μ 2 and variance σ 2 1 + σ 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 ( μ 1 + μ 2 , σ 2 1 + σ 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 = σ 1 X 1- μ 1 σ 1 + X 2- μ 2 σ 1 + ( μ 1 + μ 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#9...

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