Question

2. Let X and X be two random variables with the following joint PMF Yix 2 0 2 0 0.1 0.05 0.05 0.15 0.1 0.05 0.1 0.05 0.05 0.05 4 0.05 0.05 0.02 0.1 0.03 total 0.2 0.2 0.12 0.3 0.18 total 0.45 0.3 0.25 1 1) Find E[X] and E[Y]. (10 points) 2) What is the covariance of X and Y? (20 points) 3) Are X and Y independent? Explain. (10 points)

Answer 1

	x
y	-2	-1	0	1	2	Total
0	0.1	0.05	0.05	0.15	0.1	0.45
1	0.05	0.1	0.05	0.05	0.05	0.30
4	0.05	0.05	0.02	0.1	0.03	0.25
Total	0.20	0.20	0.12	0.30	0.18	1.00

marginal distribution of y:
y	P(y)	yP(y)	y^2P(y)
0	0.450	0.000	0.000
1	0.300	0.300	0.300
4	0.250	1.000	4.000
total	1.000	1.300	4.300
E(y)	=		1.3000
E(y^2)	=		4.3000
Var(y)=	E(y^2)-(E(y))^2=		2.6100

x	P(x)	xP(x)	x^2P(x)
-2	0.200	-0.400	0.800
-1	0.200	-0.200	0.200
0	0.120	0.000	0.000
1	0.300	0.300	0.300
2	0.180	0.360	0.720
total	1.000	0.060	2.020
E(x)	=		0.0600
E(x^2)	=		2.0200
Var(x)=σy=	E(x^2)-(E(x))^2=		2.0164

1)

from above

E(X)=0.06

E(Y) =1.30

2)

E(XY) =$\sum$xyP(x,y) =0*(-2)*0.1+0*(-1)*0.05+......+4*2*(0.18) =-0.01

Covar(x,y)=E(XY)-E(X)*E(Y)=

-0.0880

3)

No as Covariance is not equal to 0. therefore X and Y are not independent

(even if that would be we need to check for each cell if P(x,y) =P(x)*P(y) for independence.)