Answer Key for Random Variables Review: Problems and Solutions - Prof. Eric Zivot, Assignments of Economics

Download Answer Key for Random Variables Review: Problems and Solutions - Prof. Eric Zivot and more Assignments Economics in PDF only on Docsity! Answer Key for Review Problems on Random Variables 1. A random variable X has the following distribution: Row 1 Possible values of X: 0 1 2 3 4 Row 2 Probabilities .2 .1 .2 .3 .2 Row 3 Values of 3X-1: Row 4 Values of X2 a. What two sets of information are required to specify the distribution of the random variable X? Set of possible values, row 1 and corresponding probabilities, row 2. b. Compute E(X). ∑ == 2.2ii xp c. Compute V(X). ∑ −= ]))([( 2XExp ii or better =−= 22 )]([)( XExE 1.96 d. What is the probability distribution of the new random variable Y=3X-1. [Hint: see part a.] Rows 2 and 3 (after you fill it in). e. Compute E(Y) =5.6 and V(Y)=17.64. f. Find the probability distribution of the random variable X2 and compute E(X2)=6.8. Rows 2 and 4 (filled in) g. What is the value of E(9X2-6X+1). [Hint: the easy way is to use the results above and the simple theorems.] =49.0 h. What is the value of E[(X-E(X))2]? What is another name for this expression? What does it measure? =1.96 This is V(X). It measures the dispersion of the X distribution. 2. A producer has the following cost function: Output X 500 700 1000 1500 Average Cost AC 2 3.5 6 9 Price is a random variable with the probability distribution: Price P 10 12 15 20 Probabilities .5 .3 .1 .1 The producer wants to maximize expected profits. What output should he/she choose? 10.12)( )()()( )(~ = −= −= pE xACxpEE xACxp π π Expected profits are maximized at x=1000, expected profit is 6100. 3. An investor can choose between assets A1 and A2. The probability distributions of the rates of return for these assets are given below. R1 (%) 10 15 17 20 Probabilities .4 .3 .2 .1 and R2 (%) 10 15 17 20 Probabilities .3 .3 .2 .2 Economics 422: Random Variable Review 2 a. Compute the expected return for both assets. E(r1)=13.9 E(r2)=14.9) b. Compute the variance for both assets. V(r1)=12.09 V(r2)=13.29 c. Based on the expected return vs. variance criterion, which asset would you prefer and why? Asset 2 has higher expected return but also has higher variance. There is no clear choice based on mean – variance criterion. Do you see any other basis for preferring one asset to the other? Asset 2 would be preferred by anyone. You have the same chance of getting 15 or 17, a lower chance of getting 10, and a higher chance of getting 20. A2’s distribution lies to the right of A1’s. These ideas are formalized with the concept of stochastic dominance. d. Compute the expected return and variance of a portfolio that has portfolio weights 0.4 and 0.6 and if the covariance between the two returns is zero. 5.14)2(6.)1(4.)( =+= rErErE p 7188.6)0)(6)(.4(.2)29.13(6.)09.12(4.)( 22 =++=prV Note that the portfolio variance is lower than that of either asset. 4. Random variables X and Y have the following joint distribution: X Y Prob(X,Y) 0 1 .4 0 -1 .3 1 0 .2 1 1 .1 a. First convert this information into a 2-way table showing the joint X,Y probability distribution. Y values -1 0 1 Marginal prob X 0 .3 0 .4 .7 values 1 0 .2 .1 .3 Marginal prob .3 .2 .5 1.0 b. Compute E(X), E(Y), V(X), V(Y),E(XY), Cov(X,Y), and the correlation coefficient. .3 .2 .21 .76 .1 .04 .1 5. For two jointly distributed random variables X and Y (not necessarily those of problem 4), a. If V(X)=0 or V(Y)=0, what can you say about Cov(X,Y)? Explain. Covariance would be zero. b. Show, using the definition, that Cov(aX,bY)=abCov(X,Y). c. Show, using the definition, that Cov(X+Y,Z)= Cov(X,Z)+Cov(Y,Z). 6. To test your understanding of expected utility concepts, try the following problem. Will be done in class A consumer has the utility function u=U(W)=ln(W) where u represents the level of utility, U(W) represents the general form of a utility function, W is the consumer's wealth, and ln(W)