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e ha9 (1).pdf IQ I}. .0. Problem 1. Consider two random variables X, Y with the joint distribution represented by the following table: 0.3 0.1 0.2 0.2 0.1 0.1 (3.) Find their marginal distributions, E(X), E(Y), Var(X), Var(Y). (b) Are X and Y independent? (c) Find Cov(X,Y). (d) Find Cov(X + 2, Y 2X + 1), E(X2 - (Y 1)). Problem 2. Let X , Y,Z be random variables with expectations or = [1,, = 0, u: = 1, and standard deviations at = 2, 6,, = 0,, = 1. Suppose p(X, Y) = 1/2, and XJLZ, YJLZ. Find Cov(X Z, 2X + Y). Problem 3. The Central Bank of a country projects 3 possible scenarios for the country's economic development in 2020. For each scenario, the Central Bank estimates its probability and estimates the relative changes of the country's GDP and national debt if this scenario takes place. Scenario Scenario's probability National debt Negative +2% Neutral no change Positive 1% For this model, let X denote the relative change of the GDP and Y denote the relative change of the national debt. Find the correlation coefcient of X and Y. Problem 4. Let's find the variance of the hypergeometric distribution. If X has the hypergeo metric distribution, then X = X1 + . . . + X\e ha9 (1>.pdf IQ 0+ .0. (a) If the investors invests $11 mln in the stock of company X and $(1 a) mln in the stock of company B, write the formula (containing the random variables X and Y, and the constant a.) for the random variable Z, which will be equal to his prot or loss from this investment 1 year later. Assume that a. is between 0 and 1, and that a fractional amount of shares can be bought. (b) Suppose the investors splits the money equally, $0.5 mln to the stock of each company. Find the expectation and the standard deviation of Z in that case. Use Excel or other software for computations in this and the subsequent questions. (c) Find the formulas for the expectation and the standard deviation of Z as functions of the value of 0.. (d) Consider the set of all pairs (1,5,0) Where ,u = 13(2), 0 = 0(2) for different values of a between 0 and 1. Draw this set as a curve in the plane (pa, :7). Additional problems Durrett: p. 224, problems 11, 15, 16. Bertsekas, Tsitiklis: p. 124, problems 25, 26; p. 249, problems 17-19. __\\/_