Econometrics Problem Set 1 WISE, Xiamen University Spring 2014 Conceptual Questions 1. (SW 2.9) X and Y are discrete random variables with the following joint distribution: Value of X 1 5 8 14 0.02 0.17 0.02 Value of 22 30 0.05 0.10 0.15 0.05 0.03 0.15 Y 40 0.03 0.02 0.10 65 0.01 0.01 .09 That is, Pr(X = 1, Y = 14) = 0.02, and so forth. (a) Calculate the probability distribution, mean and variance of Y . (b) Calculate the probability distribution, mean and variance of Y given X = 8. (c) Calculate the covariance and correlation between X and Y . 2. (SW 2.17) Yi , i = 1, . . . , n are i.i.d. Bernoulli random variables with p = 0.4. Let Y¯ denote the sample mean. (a) Use the central limit to compute approximations for i. Pr(Y¯ ≥ 0.43) when n = 100. ii. Pr(Y¯ ≤ 0.37) when n = 400. (b) How large would n need to be to ensure that Pr(0.39 ≤ Y¯ ≤ 0.41) ≥ 0.95? (Use the central limit theorem to compute an approximate answer.) 3. (SW 2.23) Let X and Z be two independently distributed standard normal random variables, and let Y = 3X 2 + Z. (a) What is E(Y |X). (b) What is µY . (c) Show that E(XY ) = 0. 4. (SW 2.24) Suppose that Yi is distributed i.i.d. N (0, σ 2 ) for i = 1, 2, . . . , n. (a) Show that E(Yi2 /σ 2 ) = 1. ∑ (b) Show that W = σ12 ni=1 Yi2 is distributed χ2n . (c) Show that E(W ) = n (d) Show that V = √ ∑Y1 n Y2 i=2 i n−1 is distributed tn−1 . 5. Let X be a random variable uniformly distributed between a and b. (a) What is E(X). (b) What is var(X). (c) Let X and Z be two independent random variables uniformly distributed between 0 and 1, and ∫ ∞let Y = X + Z. Derive the probability distribution of Y . (Hint: Show that FY (y) = −∞ FX (y − z)fZ (z)dxdz and differentiate both sides with respect to y.) 6. Let X be a discrete random variable with the following probability distribution. x 1 2 3 P (X = x) 1/3 1/3 1/3 (a) What is F (x)? (b) What is the mean and variance of X. (c) The median of a random variable Y is the value y such that P (Y ≤ y) ≥ 1/2 and P (Y ≥ y) ≥ 1/2. What is the median of X? (d) Give an example of a random variable which does not have a unique median. Empirical Questions No empirical questions this week. Page 2