Problem%20Set%2001_2010_2011[1]

Problem%20Set%2001_2010_2011[1] - Middle East Technical...

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Middle East Technical University 2010-2011 Fall Department of Economics ECON 206 Instructor: Ozan ERUYGUR Research Assistant: Pelin AKÇAGÜN PROBLEM SET 01 EXPECTATION AND VARIANCE OPERATORS PROBLEM 1 (Exercise 4.17 from textbook) Prove Theorem 4.5. Theorem 4.5: Let c be a constant, and let g(Y), g 1 (Y), g 2 (Y), ... , g k (Y) be functions of a continuous random variable Y. Then the following results hold: 1. E(c) = c. 2. E[cg(Y)] = cE[g(Y)]. 3. E[g 1 (Y)+ g 2 (Y)+ . .. +g k (Y)] = E[g 1 (Y)]+ E[g 2 (Y)]+ . .. +E[g k (Y)]. PROBLEM 2 (Exercise 4.18 from textbook) If Y is a continuous random variable with density function f(Y), use Theorem 4.5 to prove that . 22 () ( ) [() ] VY EY σ= = 2 PROBLEM 3 (Exercise 4.20 from textbook) If Y is a continuous random variable with mean μ and variance σ 2 and a and b are constants, use Theorem 4.5 to prove the following: a. E(aY+b)=aE(Y)+b=aμ+b. b. V(aY + b) = a 2 V(Y) = a 2 σ 2 PROBLEM 4 (Exercise 4.24 from textbook) The proportion of time Y that an industrial robot is in operation during a 40-hour week is a random variable with probability density function 2, 0 1 0, yy fy elsewhere ≤≤ = a. Find E(Y) and V(Y).
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Problem%20Set%2001_2010_2011[1] - Middle East Technical...

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