Question

Let the joint pmf of X and Y be defined by x+y 32 x 1,2, y,2,3,4 (a) Find fx(x), the marginal pmf of X. b) Find fyv), the marginal pmf of Y (c) Find P(XsY. (d) Find P(Y 2x). (e) Find P(X+ Y 3) (f) Find PX s3-Y) (g) Are Xand Y independent or dependent?Why or why not? (h) Find the means and the variances of X and Y

Answer 1

a) f(x) =$\sum_{y=1}^{4}$f(x,y) = (1/32)*((x+1)+(x+2)+(x+3)+(x+4))=(1/32)*(4x+10)=(2x+5)/16 for x=1,2

b)

f(y) =$\sum_{x=1}^{2}$f(x,y) =(1/32)*((1+y)+(2+y))=(2y+3)/32 for y=1,2,3,4

c)P(X>Y)=P(X=2,Y=1)=(2+1)/32=3/32

d)P(Y=2X)=P(X=1,Y=2)+P(X=2,Y=4)=(1+2)/32+(2+4)/32=9/32

e)P(X+Y=3)=P(X=1,Y=2)+P(X=2,Y=1)=(1+2)/32+(2+1)/32 =6/32=3/16

f)

P(X<=3-Y) =P(X=1,Y=1)+P(X=1,Y=2)+P(X=2,Y=1)=(1+1)/32+(1+2)/32+(2+1)/32=8/32=1/4

g)

as f(x,y) is not equal to f(x)*f(y) ' therefore X and Y are not independent

h)

from above marginal distribution of X:

x	P(x)	xP(x)	x^2P(x)
1	7/16	0.4375	0.4375
2	9/16	1.1250	2.2500
total	1.0000	1.5625	2.6875
E(x)	=		1.5625
E(x^2)	=		2.6875
Var(x)	E(x^2)-(E(x))^2		0.2461

hence E(X) =0.4375=25/16

Var(X)=0.2461 =63/256

similarly marginal pmf of Y:

y	P(y)	yP(y)	y^2P(y)
1	5/32	0.1563	0.1563
2	7/32	0.4375	0.8750
3	9/32	0.8438	2.5313
4	11/32	1.3750	5.5000
total	1	2.8125	9.0625
E(y)	=		2.8125
E(y^2)	=		9.0625
Var(y)	E(y^2)-(E(y))^2		1.1523

E(Y)=2.8125=45/16

Var(Y)=295/256