Problem 4
Subject:StatisticsPrice: Bought3
Problem 4.49, Page 200 illustrates a method (Behboodian 1990) for defining uncorrelated bivariate random variables (X,Y). Use parts a (without proving it) to prove part c of the problem regarding o (X, Y). 10 points
Behboodian (1990) illustrates how to construct bivariate random variables that are uncorrelated but dependent. Suppose that f1, f2, 91, 92 are univariate densities with means /1, /2, $1, $2, respectively, and the bivariate random variable (X, Y ) has density (X, Y) ~afi(x)gi(y) + (1 -a) f2(2) 92(y), where 0 < a < 1 is known. (a) Show that the marginal distributions are given by fx (x) = afi(x) + (1 -a) f2(2) and fy (x) = agi(y) + (1 - a)gz(y). (b) Show that X and Y are independent if and only if [f1 (x) - f2(x)][g1(y) -92(y)] = 0. (c) Show that Cov(X, Y) = a(1-a) [#1-M2] [$1-$2], and thus explain how to construct dependent uncorrelated random variables. (d) Letting f1, f2, 91, 92 be binomial pmfs, give examples of combinations of parame- ters that lead to independent (X, Y) pairs, correlated (X, Y) pairs, and uncorre- lated but dependent (X, Y) pairs.
Hints: 1. U N Uniform(0,1), then — 111(U) ~ Exponential?). 2. If X N gamma.(a, )8), then (:X N gamma(a, 0,8), (3 > 0 3. If Xi ~ gamma(aé,,8),then 2:":1 X,; N gamma(z?=l (11;, [3), where Xhi = 1, 2, 3..., n are independent random variables.
