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(ELEC210)[2010](s)quiz~1980^_10296.pdf
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1.
a) the joint pmf of X and Y
Y
X

-r
r 22.


0
r 22


r


-r
0
0
1/8
0
0

r 22.

0
1/8
0
1/8
0


0
1/8
0
0
0
1/8

r
0
1/8
0
1/8
0


r
0
0
1/8
0
0




. so it is not independent (,)(,)ijijPXxYyPXxYy.....

Cov(X,Y)=EXY-EXEY=0 so uncorrelated
b) p(x=-r)=1/8 p(x=r)=1/4 p(x=0)=1/4 p(x=r)=1/4 p(x=r)=1/8
p(y=-r)=1/8 p(y=r)1/4 p(y=0)=1/4 p(y=r)=1/4 p(y=r)=1/8
c) P(Xr/,Yr/)=1/8 .
2
.



2 only if. Because of property 2 (for more detail lecture notes 19 page 21), we can get . Because X, Y are independent. We can get =0. 21()exp(/2)2Xfxx...
21()exp(/2)2Yfyy...
,()()(,)XYXYfxfyfxy.
.

If .if =0, then. It is easy to verify . So X Y are independent. 22,1exp(()/2)2XYfxy....




3.
a) 1122334XYXYAXYX.......................
110001100011A...........

Because and are independent and unit-variance. 1X
2X
3X
4X
1000010000100001XK.............

210121012TYXKAKA............

b) 13/41/21/41/211/21/41/23/4YK................
||4YK.

000m...........


Replace them in the following formula. n=3
1/21/211()exp{()()}(2)||2TYYnYfyxmKxmK......

c) 12,2112YYK.......
13,2002YYK.......
23,2112YYK.......

121,2/31/31/32/3YYK..........
131,1/2001/2YYK........
231,2/31/31/32/3YYK..........

12,||3YYK.
13,||4YYK.
23,||3YYK.
1200YYm.......
1300YYm.......
2300YYm.......

Replace them in the following formula. n=2




d)
Z=AY so
TZYKAKA.
,YBecauseZareindependentweneedtodiagonalizeK

Use the knowledge In linear algebra
123TYPKP..............
1230.585823.4142......

1/22/21/22/202/21/22/21/2P...............

TsoAP.
1/22/21/22/202/21/22/21/2A...............