Exercise 1

# Exercise 1 - ECON 425 Q1 EXERCISE 1 Dr AHN A random...

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ECON 425 EXERCISE 1 Dr. AHN Q1. A random variable X has the following pdf: x 0 1 2 3 4 f ( x ) b 2 b 3 b 4 b 5 b 1) What is the value of b ? Why? 2) Find the Pr( X 3). 3) Find E( x ). Q2. The joint probability distribution of X and Y is given by the following table: (For example, f(4,9) = 0.) X \ Y 1 3 9 2 1/8 1/24 1/12 4 1/4 1/4 0 6 1/8 1/24 1/12 1) Find the marginal pdfs of X and Y . 2) Find var(2 x +3 y ). Q3. Let X stand for the rate of return on a security (say, IBM) and Y the rate of return on another security (say, General Motors). Let 0.5 XY , 2 X = 4, 2 Y = 9 and corr( x , y ) = -0.8. 1) Find E [0.5 x +0.5 y ] and var[0.5 x +0.5 y ]. 2) Is it better to invest equally in the two securities (i.e., diversify) than in either security exclusively? (Hint: Investors consider both expected rate of return and risk.) Explain in detail why or why not. Q4. Let Y ~ χ 2 (5). 1) Find a such that Pr( Y > a ) = 0.05. 2) Find c such that Pr( Y < c ) = 0.9. Q5. Let the two random variables, X 1 and X 2 , are i.i.d. with N (0,1). Find Pr( X 1 2 + X 2 2 > 9.21 ). Q6. Consider the three random variables, X , Y , and Z . Assume that all of them are stochastically independent. Let X be N (0,1); Y be χ 2 (5); Z be χ 2 (4).

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Exercise 1 - ECON 425 Q1 EXERCISE 1 Dr AHN A random...

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