We have an urn with 6 red balls and 4 green balls We draw ba

We have an urn with 6 red balls and 4 green balls. We draw balls from the urn one by one without replacement, noting the order of the colors, until the urn is empty. Let X be the number of red balls in the first five draws, and Y the number of red balls in the last five draws. Compute the covariance Cov(X, Y )

Solution

E(X)=1(6/10)+2(5/45)+3(4/120)+4(3/210)+5(2/252)=0.7666

E(Y)=6(5/210)+5(4/120)+8(3/45)+9(2/10)+10(1/1)=1.3238.

E(XY)=(9/10)+(16/45)+(21/120)+(24/210)+(25/252)+(24/210)+(21/120)+(16/45)+(9/10)+(0/1)=3.178. COV(X,Y)=E(XY)-[E(X)E(Y)]=3.188-3.779=-0.591.