Question

Please help me sovle this problem

Question 5. Thejoint probability distribution of two discrete random vari- ables X and Y is given in the following table. Take e to be fixed between -1/4 and 1/4: 1/4-? 1/4+? 1/4+1 1/4-e px(a) 1/2 1/2 py(b) 1/2 1/2 (a) Take e 1/8 and compute cov(X, Y) (b) Take-18 and compute p(X, Y). (c) For which values of e is ?(X, Y) equal to-1, 0,1?

Answer 1

a) taking $\epsilon$ =1/8 ; below is joint probability distribution:

	y
x	0	1	Total
0	1/8	3/8	1/2
1	3/8	1/8	1/2
Total	1/2	1/2	1

marginal distribution of X:

x	P(x)	xP(x)	x^2P(x)
0	0.5000	0.0000	0.0000
1	0.5000	0.5000	0.5000
total	1	0.5	0.5
E(x)	=		0.5000
E(x^2)	=		0.5000
Var(x)	E(x^2)-(E(x))^2		0.2500

marginal distirbution of Y:

y	P(y)	yP(y)	y^2P(y)
0	0.5000	0.0000	0.0000
1	0.5000	0.5000	0.5000
total	1.0000	0.5000	0.5000
E(y)	=		0.5000
E(y^2)	=		0.5000
Var(y)	E(y^2)-(E(y))^2		0.2500

E(XY) = $\sum$ xyP(x,y) =(1/8)*1*1 =1/8 =0.125

hence Cov(X,Y) =E(XY)-E(X)*E(Y) =0.125-0.5*0.5 =-0.125

b)

correlation coefficient =Cov(X,Y)/(Var(X)*Var(Y))^1/2 =-0.5

c)

for $\rho$ (X,Y) =-1 ; xshould increase when Y decrease or vice versa ; therefore $\epsilon$ =1/4

$\rho$ (X,Y) =0 ; x and Y should be independent or P(X=1;Y=1) =P(X=1)*P(Y=1) ; solving whcih $\epsilon$ =0

for $\rho$ (X,Y) =1 ; xshould increase when Y increase or vice versa ; therefore $\epsilon$ =-1/4