Question

Suppose X and Y are random variables with P(X = 1) = P(X = -1) = 12; P(Y = 1) = P(Y = -1) = 1
Let c = P(X = 1 and Y = 1).
(a) Determine the joint distribution of X and Y, Cov(X, Y), and Cor(X, Y).
(b) For what value(s) of c are X and Y independent? For what value(s) of c are X and Y 100% correlated?

Answer 1

$a) P\{X=1, Y=1\}=\frac{1}{3}\\ P\{X=1, Y=-1\}=\frac{1}{6}\\ P\{X=-1, Y=1\}=\frac{1}{6}\\ P\{X=-1, Y=-1\}=\frac{1}{3}\\ COV(X,Y)=M(XY)-M(X)M(Y)\\ M(X)=M(Y)=0\\ COV(X,Y)=M(XY)\\ P\{XY=1\}=\frac{2}{3}\\ P\{XY=-1\}=\frac{1}{3}\\ M(XY)=\frac{1}{3}\\ COV(X,Y)=\frac{1}{3}\\ Cor(X,Y)=\frac{COV(X,Y)}{\sqrt{D(X)D(Y)}}\\ D(X)=M(X^2)-(M(X))^2\\ M(X)=0\\ M(X^2)=1\\ D(X)=1\\ D(Y)=1\\ Cor(X,Y)=COV(X,Y)\\ Cor(X,Y)=\frac{1}{3}\\ b)1)c=P\{X=1, Y=1\}=P\{X=1\}P\{Y=1\}=\frac{1}{4}\\ X \text{ and } Y \text{ are independent for }c=1/4.\\ 2)\text{Let }Cor(X,Y)=1\\ Cor(X,Y)=\frac{COV(X,Y)}{\sqrt{D(X)D(Y)}}\\ Cor(X,Y)=COV(X,Y)\\ COV(X,Y)=M(XY)-M(X)M(Y)\\ M(XY)=1\\ \text{Hence } P\{X=1,Y=1\}+P\{X=-1,Y=-1\}=1,\\ P\{X=1,Y=-1\}+P\{X=-1,Y=1\}=0.\\ P\{X=1,Y=1\}=\frac{1}{2}\\ P\{X=-1,Y=-1\}=\frac{1}{2}\\ P\{X=1,Y=-1\}=0\\ P\{X=-1,Y=1\}=0\\ \text{So for } c=\frac{1}{2} \text{ X and Y are 100 percent correlated}.$