Question

3 The table shows a joint pdf. Find the covariance and the correlation coefficient for X and Y Cov(X,Y)= 1 2 4 3 0.125 0 0 4 0.25 0 0 '50 0. 50 6 0 0 0.125

Answer 1

x	y	f(x,y)	x*f(x,y)	y*f(x,y)	x^2f(x,y)	y^2f(x,y)	xy*f(x,y)
1	3	0.125	0.125	0.375	0.125	1.125	0.375
2	3	0	0	0	0	0	0
4	3	0	0	0	0	0	0
1	4	0.25	0.25	1	0.25	4	1
2	4	0	0	0	0	0	0
4	4	0	0	0	0	0	0
1	5	0	0	0	0	0	0
2	5	0.5	1	2.5	2	12.5	5
4	5	0	0	0	0	0	0
1	6	0	0	0	0	0	0
2	6	0	0	0	0	0	0
4	6	0.125	0.5	0.75	2	4.5	3
	Total	1	1.875	4.625	4.375	22.125	9.375
	E(X)=ΣxP(x,y)=					1.875
	E(X2)=Σx2P(x,y)=					4.375
	E(Y)=ΣyP(x,y)=					4.625
	E(Y2)=Σy2P(x,y)=					22.125
	Var(X)=E(X2)-(E(X))2=					0.86
	Var(Y)=E(Y2)-(E(Y))2=					0.73
	E(XY)=ΣxyP(x,y)=					9.375

from above"

Cov(X,Y)=E(XY)-E(X)*E(Y)=					0.7031
Correlation =ρxy =Cov(X,Y)/sqrt(Var(X)*Var(Y))=					0.8851