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4.9 a) For x > 0, y > 0
[:1: [Y 2 / 2 FXY(x, y) = Jo Jo axe-a:1:2bye-by/2dxdy
= (1 - e-ax2/2)(1- e-by2/2)
b) P[X>y] = l°O1xaxe-ax2/2bye-by2/2dydX
[00 2 2 = Jo axe-a:1:/2(1 - e-b:1:/2)dx
a = 1--
a+b
c) Fx(x) = lim Fx(x,y) = 1- e-a:1:2/2 x> 0
y-+oo
- fx(x) = 1;Fx(x) = axe-ax2/2 x > 0
Similarly
fy (y) = bye-by2/
100 (-
fx(x) = xe-xe-xYdy = xe-:1: -e-XY ) 00 = e-:1:
4.13 0 x 0
f ( ) - 100 -:1:(l+Y) d - e-x(l+y)((1 + y)x - 1)
y y - xe x - 1 00
0 (1 + y)2 0
4.17 a) P[X = i,Y ~ y] = P[Y ~ ylX = i]P[X = i]
If X = 1:
1 1 [y-l a
P[Y ~ ylX = l]P[X = 1] = P[N + 1 ~ y]"2= "2J-oo 2e-a/zldz
{ !ea(y-l) y < 1 = 1(2 - e-a(y-l)) y > 1
X = -1 is obtained in similar fashion:
{ !ea(y+l) y < - P[Y ~ ylX = l]P[X = -1] = 1(2 - e-a(y+l)) y ~ -
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b) Fy(y) = pry ~ ylX =]P[X = 1]+ pry ~ ylX = -1]P[X = -1]
= P[N + 1 ~ y]2 + P[N - 1 ~ y]
d Id Id fy(y) = dYFy(y) = 2dYFN(Y- 1) + 2dYFN(Y+ 1)
= 2fN(y - 1) + 2fN(y + 1)
~_. ///II~~~-
I ;..y
0
///'/""""I :~---//'~"'I I ~
-1 0 1 Y
c) P[X = 11Y > 0] = pry > OIX = I]P[X = 1] = 1 - ~e-a
pry > 0] 2P[Y > 0]
P[X = -IIY > 0] = pry > OIX = -1]P[X = -1] = !(1 + !e-a)
pry > 0] 2P[Y > 0]
1(1 -a)
P[X = 11Y> 0] - P[X = -11Y > 0] = 2 - e > 0
2P[Y > 0]
~ X = 1 is more likely
4.24 a) P[X2 < !, IY - 11< !] = P[X2 < !]P[IY - 11 < !]
= P[X < 1 ]P[Y > 1]^ =^1
~ 2 2~
b) P[Xj2 < I,Y > 0] = P[X < 2]P[Y > 0] = 1.
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b) If p = 0 then
x
P[XY > 0] = P[X and Y have same sign]
= JJ ++ fxy(x,y)dxdy+JJ -- fXy(x,y)dxdy
quadrant quadrant
.,~~ =^100 fx(x)dx^100 fy(y)dy^ +^10 fx(x)dx^10 fy(y)dy
~ 0 0 -00 -
i:
;but
100 1 100 -tC/
e-(x-mlr/20"~dx= ~dt = Q (-~ )
0 V2;0"1 -~ V2; 0"
1 and similarly for other integrals, thus
\ P[XY > 0] = Q (-~) Q (-~) + (1 - Q (-~)) (1- Q (-~))
.;
, ,,-~,;,
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