Question

Let X and Y have the following joint distribution:

X/Y	0	1	2
0	5/50	8/50	1/50
2	10/50	1/50	5/50
4	10/50	10/50	0

Further, suppose σx = √(1664/625), σy = √(3111/2500)

a) Find Cov(X,Y)

b) Find p(X,Y)

c) Find Cov(1-X, 10+Y)

d) p(1-X, 10+Y), Hint: use c and find Var[1-X], Var[10+Y]

Answer 1

for above

	y
x	0	1	2	Total
0	1/10	4/25	1/50	0.2800
2	1/5	1/50	1/10	0.3200
4	1/5	1/5	0	0.4000
Total	0.5000	0.3800	0.1200	1.0000

a)

marginal distribution of X:

x	P(x)	xP(x)	x^2P(x)
0	7/25	0.0000	0.0000
2	8/25	0.6400	1.2800
4	2/5	1.6000	6.4000
total	1	2.24	7.68
E(x)	=		2.24
E(x^2)	=		7.68
Var(x)	E(x^2)-(E(x))^2		2.6624

marginal distribution of Y:

y	P(y)	yP(y)	y^2P(y)
0	1/2	0.0000	0.0000
1	19/50	0.3800	0.3800
2	3/25	0.2400	0.4800
total	1.0000	0.6200	0.8600
E(y)	=		0.6200
E(y^2)	=		0.8600
Var(y)	E(y^2)-(E(y))^2		0.4756

E(XY)=xyP(x,y)=1.24

a)

Cov(X,Y) =E(XY)-E(X)*E(Y)=-0.1488

b)

p(X,Y) =Cov(X,Y)/( σx * σy)=-0.13223

c)

as Cov(aX+b,cY+d)=ac*Cov(X,Y)

hence Cov(1-X, 10+Y) =-1*1*Cov*X,Y)=0.1488

d)

p(1-X, 10+Y) = Cov(1-X, 10+Y) /( σx * σy)=0.13223