Question

1. Let the joint probability (mass) function of X and Y be given by the following: Value of X -1 -1 3/8 1/8 Value of Y1 1/8 3/8 (a) Determine the marginal (b) Determine the conditional distribution of X given Y (c) Are they independent? d) Compute E(X), Var(X), E(Y) and Var(Y). (e) Compute PXY <0) and Ptmax(X,Y) > 0 (f) Compute Elmax(X, Y)] and E(XY) (g) Compute Cov(X,Y) and Corr(X, Y) 1

Answer 1

	x
y	-1	1	Total
-1	3/8	1/8	1/2
1	1/8	3/8	1/2
Total	1/2	1/2	1

a)

marginal distribution of X:

x	P(x)
-1	1/2
1	1/2

marginal distribution of Y:

y	P(y)
-1	1/2
1	1/2

b)

below is condition distribution of X given Y=1

P(X=-1|Y=1)=(1/8)/(1/2)=1/4

P(X=1|Y=1)=(3/8)/(1/2)=3/4

c)

No as P(X|Y=1) is different for X =-1,1 therefore X and Y are not independent

d)

x	P(x)	xP(x)	x^2P(x)
-1	1/2	-0.5000	0.5000
1	1/2	0.5000	0.5000
total	1.0000	0.0000	1.0000
E(x)	=		0.0000
E(x^2)	=		1.0000
Var(x)	E(x^2)-(E(x))^2		1.0000

from above E(X)=0.00

Var(X)=1.00

y	P(y)	yP(y)	y^2P(y)
-1	1/2	-0.5000	0.5000
1	1/2	0.5000	0.5000
total	1	0	1
E(y)	=		0.0000
E(y^2)	=		1.0000
Var(y)	E(y^2)-(E(y))^2		1.0000

E(Y)=0.00

Var(Y)=1.00

e)

P(XY<0)=P(X=-1,Y=1)+P(X=1,Y=-1)=1/8+1/8=1/4

P(max(X,Y)>0)=1-P(max(X,Y)<0)=1-P(X=-1,Y=-1)=1-3/8=5/8

f)

E(max(X,Y))=$\sum$max(X,Y)*P(x,y)=(3/8)*max(-1,-1)+(1/8)*(1,-1)+(3/8)*max(1,1)+(1/8)*(-1,1)

=2/8=1/4

EXY)=$\sum$xy*P(x,y)=0.5

g)

Cov(X,Y)=E(XY)-E(X)*E(Y)=0.50

Corr(X,Y)= Cov(X,Y)/sqrt(Var(X)*Var(Y))=0.50